Bessel and Related Functions: Volume 1 Theoretical Aspects 9783110681642, 9783110681574

Table of contents :
Preface
Contents
1. Introduction
2. Properties of the Bessel and Related Functions
3. Differentiation and Integration with Respect to the Order of the Bessel and Related Functions
4. Mathematical Operations with Respect to the Order of the Bessel and Related Functions – Integration, Differentiation, Series and Limits
References
Appendix A. The Bessel and Related Functions Integrals
Appendix B. Limits Representing Special Functions
Appendix C. Notation and Definitions of Special Functions
Index

Citation preview

Alexander Apelblat Bessel and Related Functions

Also of interest Bessel and Related Functions. Mathematical Operations with Respect to the Order Volume : Numerical Results Alexander Apelblat,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----, Set-ISBN ---- Combinatorial Functional Equations. Basic Theory Yanpei Liu,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Combinatorial Functional Equations. Advanced Theory Yanpei Liu,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Tensor Numerical Methods in Scientific Computing Series: Radon Series on Computational and Applied Mathematics  Boris N. Khoromskij,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Fractional-Order Equations and Inclusions Series: Radon Series on Computational and Applied Mathematics  Michal Fečkan, JinRong Wang, Michal Pospíšil,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----

Alexander Apelblat

Bessel and Related Functions Mathematical Operations with Respect to the Order Volume 1: Theoretical Aspects

Author Prof. Dr. Alexander Apelblat Ben-Gurion University of the Negev Department of Chemical Engineering Beer Sheva, Israel [email protected]

ISBN 978-3-11-068157-4 e-ISBN (PDF) 978-3-11-068164-2 e-ISBN (EPUB) 978-3-11-068181-9 Library of Congress Control Number: 2020933594 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: Anon. (2010). Tinted detail of Friedrich Wilhelm Bessel [digital image]. http://www.dspace.cam.ac.uk/handle/1810/224200. University of Cambridge, Institute of Astronomy Library. CC BY 4.0. Licence: https://creativecommons.org/licenses/by/4.0/ Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

To Ira and Yoram

Preface I encountered infinite series of the Bessel functions of integer orders rather early, when in the seventies of the previous century I did my PhD thesis. I solved the Navier–Stokes and the Darcy partial differential equations in cylindrical geometry. These equations expressed the physical situation which included the blood flow in fine capillaries with permeable walls and the plasma movement into surrounding tissue space. This was a mathematical modeling of the capillary-tissue fluid exchange in a human body (microcirculation). For the first time, a detailed distribution of velocities and pressures in tissue regions, induced by the blood flow in capillaries, was determined. In the solution of this mathematical problem, an important part plied series of the Bessel functions. The sums of infinite series of the Bessel functions of integer orders were calculated by using a central computer, with a help of punched cards, and by employing old versions of Fortran programs [1]. In the next step in my engineering career, this time in yearly eighties, I met with the Bessel and the Airy functions in the context of the heat and mass transfer coupled with homogeneous and heterogeneous chemical reactions of the first order. Analytical solutions were proposed for various types of flows and, in different geometric configurations. Analogous problems of the heat and mass transfer, but taking also into account the axial diffusions were also considered. These physical problems were solved by applying the Laplace transform methods, and they included infinite integrals with integrands containing the Bessel and the Airy functions [2, 3]. However, my most important involvement with the Bessel functions took place in 1985 when together with late Naftali Krivitsky we investigated the integral representations of integrals and derivatives with respect to the order of the Bessel functions, the integral Bessel function and the Anger function [4]. In a similar way, I later examined the mathematical properties of the Struve functions and the Kelvin functions [5, 6]. There is a huge literature devoted to various properties of the Bessel or related functions and the subject continue to be active nowadays. However, one topic, is much less in the center of interest of mathematicians or theoretical physicists, the mathematical operations with respect to the order of the Bessel functions. This is rather strange situation, because the first investigations on this subject are already mentioned in the fundamental Watson treatise on the Bessel functions [7], which was published at beginning of previous century. Since these few results and not many derived later are dispersed in mathematical literature, it seems desirable to present them in a more systematic, much visible form. Personally, I was intrigued by the mathematical operations with respect to the order or with respect to the parameters of other special functions, and for many years I had an intention to write a small book on the subject which would be devoted to the Bessel and related functions. Over many years I collected material for https://doi.org/10.1515/9783110681642-202

VIII

Preface

such project, but the present two volume book is a result of my intense efforts during the last three years. During the last two decades appeared some new results dealing with various properties of the Bessel functions, they are not included in standard monographs or books on the subject, and therefore I decided to place them also in this book. Some less known in the literature topics when the order plays role of independent variable are also incorporated. To make reading of presented material easy, in consistent form, one chapter is devoted entirely to the basic properties of the Bessel and related functions. This book includes primarily material connected with mathematical operations with the respect to the order of the Bessel and related functions, but it also contains many formulas, equations and mathematical techniques which will be of interest in different scientific disciplines, and in solving various technical problems. Evidently, this book is directed first of all to mathematicians who have interest in properties of special functions, but it is also relevant for researchers and graduate students in mathematics, mathematical physics, physical chemistry and some engineering disciplines. It is also expected that it will stimulate new research and more interest on the subject which is considerably less known in mathematical literature or applied in mathematical practice. This book is divided into two parts (Part 1 – Theoretical Aspects and Part 2 – Numerical Results), and it provides the up-to-date survey of mathematical operations with regard to the order of the Bessel and related functions. Part 1 contains four Chapters, References and tree Appendices. Introduction which is considered as Chapter 1, gives a short historical look at the available on the subject literature. In Chapter 2, are presented briefly the basic properties of the Bessel and related functions. They are given for convenience of readers, but also because these formulas are intensively used throughout this book. The presentation of material is arranged in such way, to make a location of desired formulas easy. In Chapter 2, large section is devoted to solutions of the second or higher order differential equations which can be expressed in terms of the Bessel functions. Some of these solutions were derived by Hayek in terms of the Bessel-Clifford functions [8], but here they are expressed in terms of the Bessel functions. Chapter 3 includes a survey of investigations devoted to mathematical operations with the order of Bessel and related functions. Besides, a new material is also added, which includes integrals and higher order derivatives with respect to the order and other related topics. All presented functional expressions in Chapter 3 are coming from the literature, and therefore they are given without detailed derivations or proofs. Most of material in Chapter 3 was compiled from handbooks dealing with special functions [9–21], monographs dedicated to the Bessel functions [7, 22–26], tabulations of series, infinite integrals and integral transforms [15, 27–41], and few important contributions published in physical and mathematical journals [42–57]. Chapter 4 is entirely devoted to mathematical operations dealing with the order of Bessel and related functions. The order as an independent

Preface

IX

variable participates in differentiation and integration operations, in the location and evaluation of order zeros and interrelations between arguments and orders. Different approaches and procedures are illustrated, and advantages associated with operational techniques are demonstrated. Two sections in Chapter 4 are devoted to zeros of the Bessel functions which are considered as a function of the order, and to zeros of the Bessel and related functions with respect to the argument when the order is changed. The last subject has been popular during many decades, as the special case when the Bessel functions have the same argument and order. The benefit to use the shifted Dirac delta function which is defined as a limit, when the order of the Bessel function is variable, is also thoroughly illustrated in Chapter 4. Appendix A contains miscellaneous integrals of the Bessel and related functions compiled from the recent literature, but not included in the main tabulations of integrals of special functions. The integral representations of special functions as the limits of infinite integrals are given in Appendix B. Notation and definitions of special functions is presented in Appendix C. Part 2 contains four Chapters where are reported numerical results which include the first, second and third derivatives with respect to the order at fixed values of arguments, and in some cases the involved functions. They are presented in tables and in a graphical form. These derivatives were calculated for the first time for the Bessel and related functions. I appreciate very much Professor Marija Bĕster-Rogaĕ from Department of Physical Chemistry, Ljubljana University, Ljubljana, Slovenia and Professor Francesco Mainardi, Department of Physics and Astronomy, University of Bologna, Bologna, Italy who helped me when I had difficulty to obtain publications associated with discussed here topics. I am also greatly indebted to Professor Larry Glasser from Department of Physics and Mathematics, Clarkson University, Potsdam, N.Y. who derived a general formula that permitted to express the Bessel – Clifford functions in terms of the Bessel functions. All numerical calculations in this book were performed by using MATHEMATICA program. I am thankful to Dr. Juan Luis GonzálesSantander, Valencia, Spain who introduced me to this program. Time spent in the preparation of this book was evidently taken from my family, and therefore, I am truly grateful to my wife Ira and my son Yoram for their continuous encouragement, profound understanding that I am not always available at home, and for so many years of constant assistance and patience. Alexander Apelblat January 2017 – August 2019.

Contents Preface

VII

1

Introduction

1

2

Properties of the Bessel and Related Functions 5 2.1 Definitions and Notations of the Bessel and Related Functions and Associated Differential Equations 5 2.2 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Bessel Jν(z) and Yν(z) and the Hankel Hν(1)(z) and 12 Hν(2)(z) functions 2.3 Interrelations, Recurrence Relations, Differential and Integral 14 Formulas for the Modified Bessel Iν(z) and Kν(z) Functions 2.4 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Struve Hν(z) and the Modified Struve Lν(z) Functions 16 2.5 Interrelations, Recurrence Relations and Differential Formulas for 18 the Lommel sµ,ν(z) and Sµ,ν(z) Functions 2.6 Interrelations, Recurrence Relations, Differential and Integral 20 Formulas for the Anger Jν(z) and the Weber Eν(z) Functions 2.7 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Kelvin berν(z), beiν(z), the Modified Kelvin kerν(z), 21 keiν(z) and herν(z), heiν(z) Functions 2.8 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Bessel Integral Functions 22 2.9 Series Expansions of the Bessel and Related Functions 23 2.10 Bessel and Related Functions of Half Odd Integer Order 27 2.11 Bessel and Related Functions Expressed in Terms of Special Functions 31 2.12 Integral Representation of the Bessel and Related Functions 33 2.13 Differential Equations Reducible to the Bessel Differential Equations 43

3

Differentiation and Integration with Respect to the Order of the Bessel and Related Functions 78 3.1 First Derivatives of the Bessel and Related Functions with Respect to the Order ν: Unrestricted Values of the Order ν 78 3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order ν: Particular Values of the Order ν 86 3.3 Derivatives with Respect to the Order ν of the Bessel and Related Functions Based on Integral Representations 101

XII

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3.4

4

Higher Order Derivatives of the Bessel and Related Functions with Respect to the Order ν 119

Mathematical Operations with Respect to the Order of the Bessel and Related Functions – Integration, Differentiation, Series and Limits 134 4.1 Integration with Respect to the Order of the Bessel and Related Functions 134 4.2 Integration and Differentiation with Respect to the Order of the Bessel Function of the First Kind by Applying the Laplace Transform Approach 142 4.3 Integration and Differentiation with Respect to the Order of the Modified Bessel Function of the First Kind by Applying the Laplace Transform Approach 154 4.4 Differentiation with Respect to the Order of the Anger Function and the Weber Function by Applying the Laplace Transform Approach 162 4.5 Integration and Differentiation with Respect to the Order of the Struve Functions by Applying the Laplace Transform Approach 168 4.6 Differentiation with Respect to the Order of the Kelvin Functions by Applying the Laplace Transform Approach 176 4.7 Differentiation with Respect to the Order of the Integral Bessel Functions by Applying the Laplace Transform Approach 179 4.8 Zeros of the Bessel and Related Functions As a Function of Their Order 183 4.9 Zeros of the Bessel Functions with Respect to the Argument As a Function of Their Order 193 4.10 Bessel Functions of Equal or Nearly Equal Order and Argument 211 4.11 The Asymptotic Limit of the Bessel Function ν Jν(νx) Expressed as the Shifted Dirac Function δ(x – 1). Evaluation of Integral Representations of Elementary Functions, Special Functions, Mathematical Constants, Integral Transforms, Asymptotic Relations, Integrals and Limits of Functional Series 216

References

241

Appendix A The Bessel and Related Functions Integrals Appendix B Limits Representing Special Functions

279

Appendix C Notation and Definitions of Special Functions Index

321

249

315

1 Introduction Starting from the end of the seventeenth century, in the context of solution of the Riccati-type differential equations, first elements of the present-day Bessel functions appeared. They are associated with different mathematical problems solved by the Bernoulli brothers, Leibniz, Euler, Lagrange, Laplace, Fourier, Poisson and others. These functions are named after the German astronomer Friedrich Wilhelm Bessel (1784–1846) who during the period 1822–1824 had shown that the radius vector in planetary motion can be expressed by the expansion similar to that established for the Bessel functions, he also obtained many other results associated with them. During the second part of the nineteenth century, a very important progress had been achieved in the theory of the Bessel functions and more generally of the cylindrical functions. This is a result of investigations performed by famous mathematicians Jacobi, Schlömilch, Lommel, Neumann, Sonine, Hankel, Struve and Weber. Significant contributions can also be attributed to Rayleigh, Lamb, Duhamel, Stokes and Thomson (Lord Kelvin), who solved physical and engineering problems, usually in the field of hydrodynamics, heat transfer, strength of materials and electricity. In these investigations, basic properties of the Bessel functions, such as representations of these functions by series expansions, recurrence formulas, asymptotic expansions and zeroes of the Bessel functions have been established. They also included definite and indefinite integrals with integrands having the Bessel functions, the integral representations of the Bessel functions, differential equations solved in terms of the Bessel functions, tabulations of the Bessel and related auxiliary functions and many other results. There is a little doubt that from special functions, the Bessel functions are the most frequently investigated and applied in mathematical, physical and engineering studies. All knowledge about the Bessel functions until the year 1922 is compiled and discussed by G.N. Watson in his monumental monograph A Treatise on the Theory of Bessel Functions [7]. Evidently, this heavy book of 804 pages became some kind of “bible” to anyone who was or is involved with the study of the Bessel functions. Similarly, as with other special functions, with an increasing modernization of society in the twentieth century, expressed by needs to solve many problems in mathematical physics and in various branches of engineering, the investigations dealing with the Bessel functions flourished. The Bessel functions are essential tools in electrodynamics, field theory, quantum and molecular physics, laser optics, pattern recognition and many other scientific and industrial disciplines. The most important results of performing investigations are summarized in a number of books that are entirely devoted to the Bessel functions and their associated integrals. Besides, in mathematical handbooks that are dedicated to properties of special functions, [7, 9, 10, 12, 17, 18, 20–28, 33–35, 39–41, 58, 59], special chapters are devoted to the Bessel and related functions. In addition, a continuous https://doi.org/10.1515/9783110681642-001

2

1 Introduction

study of the Bessel functions’ properties produced a huge number of papers that are dispersed in mathematical, physical and engineering journals. From an enormous spectrum of discussed subjects that are associated with the Bessel functions, one topic found only limited interest in mathematical literature. The Bessel functions are defined as analytic functions of two variables, their argument and order. Almost all efforts have been directed to properties of the Bessel functions connected with arguments at fixed orders. On the contrary, from a concise historic survey given here, it is clear that a number of investigations directed towards operations such as differentiation or integration with respect to the order of the Bessel functions is very small, and they received little attention in the mathematical literature. The Bessel functions of real or complex order are mentioned many times in the Watson monograph, but usually the order is not considered as an independent variable. However, there are few exceptions, the first case is when the asymptotic expansion of the Bessel functions of large order is discussed; and in the second case, the Bessel functions of equal order and argument are examined [7]. There are also few occasions when the first derivatives with respect to the order are used to obtain the second solution of the Bessel differential equation and this is also extended to various products of the Bessel functions. These results are derived by direct differentiations with regard to the order and by using basic properties of the Bessel functions. There is an example of considerable beauty: it came from the investigation of Ramanujan, his very famous integral published in 1920, where the integration variable ν is the order of Bessel functions [7]: ∞ ð

Jμ + ν ðaÞ Jλ − ν ðaÞdν = Jλ + μ ð2 aÞ −∞

a>0

;

(1:1:1)

Reðλ + μÞ > 1

Considering the operations with regard to the order, an allied subject – the Kontorovich–Lebedev integral transform – should also be noted [18]. This transformation has a large number of applications in mathematical physics, but it is far from the material discussed here; hence, it is beyond the scope of this book. In view of its applicable character, it should be treated separately. The history of investigations directed towards operations with regard to the order of the Bessel functions is rather short, and therefore probably all of them are mentioned here. They include four distinct subjects, the first rather large topic is dedicated to the representation of derivatives of Bessel functions with respect to the order, usually in terms of series or other functions [9, 18]. The second group of investigations is dealing with numerical calculations, tabulations of the Bessel functions as a function of the order, and determination of the first and second derivatives for particular values of the order [42, 43, 49, 55]. Few studies of the third subject are devoted to locations of the order of zeros at fixed arguments of the Bessel functions [60–72]. And

1 Introduction

3

finally, by applying the Laplace transform techniques, the fourth topic deals with an integration when the order serves as an integration variable [4]. It is worthwhile to note that Müller [73], G. Petiau [23] and others introduced a special, simple notation for differentiation with regard to the order of Bessel functions as Jν* ðzÞ. It replaces the explicit form of differentiation ∂Jν(z)/∂ν, but both notations are simultaneously used in the literature. As mentioned earlier, only few results dealing with derivatives of the Bessel functions with respect to the order can be found in the Watson treatise [7]. The first derivatives appeared in the context of definitions of the Bessel functions with an integer order and when orders ν and arguments z are equal or nearly equal. Watson also reports values of functions and their first derivatives for orders and arguments being integers. From many available tables in his book, it is possible to obtain the change caused when the order is altered at fixed value of an argument (e.g. see Mitra [74, 75]). In particular cases of the order, v = ± 1/2 and v = ± 3/2, Airey in 1928 [42] tabulated values of ∂Jν(x)/∂ν. In 1935, he performed [43] a series of more extensive calculations, which included the second derivatives ∂2Jν(x)/∂ν² and the functional dependence of the Bessel functions at fixed values of the order v = 0, v = ± 1/2 and v = 1. Based on the recurrening relations of the Bessel functions, Müller [73], in 1940 reported general formulas for the first derivatives of the Bessel functions with respect to the order. In 1958, Oberhettinger [56] established expressions for the first derivatives at v = ± (1/2 + n), n = 0, 1, 2, 3, . . . in terms of the exponential sine and cosine integrals. In 1960, Lee and Radosevich [55], in the 1 ≤ z ≤ 15 range, tabulated values of ∂Jν/∂ν for v = n + α; n = 1, 2, 3 and 4 and α = 1/4, 1/3, 2/3 and 3/4. In 1963, Erber and Gordon [49] calculated the first derivatives with respect to the order of the modified Bessel function ∂Iν(x)/∂ν for v = ± 1/3 in the 0 ≤ z ≤ 2.5 range and derived a number of asymptotic expressions for various ranges of argument z. Order derivatives for integer values of ν at ν = 1/2 were also expressed in terms of the sums of Bessel functions [18]. In 1977, Wienke [57] gave, explicitly, the first and second order derivatives in terms of the Neumann function expansion. In 2005, after a long interruption, Brychkov and Geddes [44] derived in the closed form, the formulas for the first derivatives of Jν(z), Yν(z), Iν(z) and Kν(z) for v = ± 1/2 ± n and n = 0, 1, 2, 3, . . . . They extended their results to the Struve functions Hν(z) and Lν(z) for v = ± 1/2 ± n and to the integral Bessel functions Jiν(z), Yiν(z) and Kiν(z) for v = ± n. Some results are expressed in the form of integral representations using the Meijer G-function and the generalized hypergeometric functions. In 2007, Brychkov [45] derived the first derivatives with respect to the order of the Weber E(z) and the Anger J(z) functions for v = ± n. This work was continued in 2016 by considering the Kelvin functions berν(z), beiν(z), kerν(z) and keiν(z), and the second and third derivatives of the Bessel functions Jν(z), Yν(z), Iν(z) and Kν(z). During the same period, few new results associated with the order derivatives of the Bessel functions were presented by Dunster [48]. In 1954, Cooke, [46] in a small note, gave three new integrals of the Ramanujan type and his work continued during the period 1993–1997 by Fényes [50–52, 76]. Watson, [57, 77] in 1916, started to

4

1 Introduction

investigate behaviour of the Bessel functions of equal order ν and argument z. The same topic was considered later in 1992 by Lorch [78]. In the context of the theory of scattering and transmission of electromagnetic waves, Magnus and Kotin [68] in 1960, Keller et al [66] in 1963, Cohen [64] in 1964, Streifer [70–72] and Cochran [65] in 1965 determined ν – values that satisfy the equations of the type Jν(ν) = 0 and ∂Jν(ν)/∂ν = 0. Similar problems were discussed by Franz and his co-workers [79, 80] during the period 1954–1957. They presented solutions in the form of Green functions applicable to cylinder and sphere geometries. However, the first investigation on a similar mathematical problem was performed much earlier, during the period 1918–1919, by Watson [57]. He considered the transmission of electrical waves around the Earth. The Coulomb work [47] from 1936 also belongs to this group of investigations. An application of the Laplace transformation techniques was initiated by van der Pol [81, 82] in 1929 by using the convolution theorem to evaluate integrals with respect to the order. A more extensive use of the Laplace transform is demonstrated in 1985 by the present author and Kravitsky [4]. They established integral representations of derivatives and integrals with respect to the order of the Bessel, Anger and Bessel integral functions. The Laplace transformation technique was also applied in the case of the Struve functions in 1989 [5] and the Kelvin functions in 1991 [6]. During the 1998–2004 period, the present author showed that the Lamborn [83] expression for the shifted Dirac delta function δðz − 1Þ = lim ½Jν ðν zÞ ν!∞

(1:1:2)

permits to evaluate integral representations of elementary and special functions, asymptotic relations, various integrals and limits of series [29, 84, 85].

2 Properties of the Bessel and Related Functions 2.1 Definitions and Notations of the Bessel and Related Functions and Associated Differential Equations The basic properties of Bessel and related functions are compiled here from books devoted to special mathematical functions and particularly from these dedicated solely to the Bessel functions Jν(z) – Bessel function of the first kind of order ν Yν(z) – Bessel function of the second kind of order ν (also called as the Neumann or the Weber functions) The order ν and argument z can be arbitrary complex or a real number, but Jν ðzÞ cosðπ νÞ− J−ν ðzÞ ; ν ≠ 0, 1, 2, 3, . . . sinðπ νÞ   ∂ Jν ðzÞ ∂ J−ν ðzÞ n = 0, 1, 2, 3, . . . − ð− 1Þn Yν ðzÞ = lim ½Jν ðzÞ = ν!n ∂ν ∂ν ν=n

Yν ðzÞ =

(2:1:1) (2:1:2)

Bessel functions are solutions of the Bessel differential equation d2 wðzÞ d wðzÞ +z + ðz2 − ν2 ÞwðzÞ = 0 d z2 dz     ν2 wðzÞ = 0 z wðzÞ′ ′ + z − z z2

(2:1:3) (2:1:4)

where wðzÞ = A Jν ðzÞ + B Yν ðzÞ ≡ Cν ðzÞ

(2:1:5)

and A and B are integration constants. Iν(z) – modified Bessel function of the first kind of order ν Kν(z) – modified Bessel function of the second kind of order ν (McDonald functions) The modified Bessel functions are solutions of the following Bessel differential equation: z2

d2 wðzÞ d wðzÞ +z − ðz2 + ν2 ÞwðzÞ = 0 d z2 dz

(2:1:6)

where wðzÞ = A Iν ðzÞ + B Kν ðzÞ ≡ Zν ðzÞ https://doi.org/10.1515/9783110681642-002

(2:1:7)

6

2 Properties of the Bessel and Related Functions

Hνð1Þ ðzÞ – Hankel function of the first kind of order ν Hνð2Þ ðzÞ– Hankel function of the second kind of order ν Hνð1Þ ðzÞ = Jν ðzÞ + i Yν ðzÞ

(2:1:8)

Hνð2Þ ðzÞ = Jν ðzÞ− i Yν ðzÞ

(2:1:9)

Hν(z) – Struve functions of order ν Lν(z) – modified Struve functions of order ν The Struve functions of order ν are particular solutions of the following differential equation z ν + 1 4 2 d wðzÞ d wðzÞ +z z2 + ðz2 − ν2 ÞwðzÞ = pffiffiffi 2 d z2 dz π Γ ν + 21

(2:1:10)

sµ,ν(z) – Lommel functions of the order µ and ν provided that μ ± ν ≠ − 1, − 2, − 3, . . . Sµ,ν(z) – Lommel functions of the order µ and ν with arbitrary values µ and ν These functions are particular solutions of the following differential equation z2

d2 wðzÞ dwðzÞ +z + ðz2 − ν2 ÞwðzÞ = zμ d z2 dz

(2:1:11)

Jν(z) – Anger functions of the order ν The Anger function satisfies the following differential equation: z2

 z − ν d2 wðzÞ d wðzÞ +z + ðz2 − ν2 ÞwðzÞ = sinðπ νÞ 2 dz dz π

(2:1:12)

and is also defined by the finite integral ðπ 1 cosðν t − z sin tÞdt J ν ðzÞ = π 0

Eν(z) – Weber functions of the order ν The Weber function satisfies the following differential equation

(2:1:13)

7

2.1 Definitions and Notations of the Bessel and Related Functions

z2

d2 wðzÞ d wðzÞ ½z + ν + ðz − νÞ cosðπ νÞ +z + ðz2 − ν2 ÞwðzÞ = − d z2 dz π

(2:1:14)

and is also defined by the integral Eν ðzÞ =

ðπ 1 sinðν t − z sin tÞdt π

(2:1:15)

0

berν(z) – Kelvin function, the real part of the Bessel function Jν(z) of order ν and of the argument i3/2z beiν(z) – Kelvin function, the imaginary part of the Bessel function Jν(z) of order ν and of the argument i-3/2z The order ν is a real, non-negative number and ber0 ðzÞ = berðzÞ

; bei0 ðzÞ = beiðzÞ

(2:1:16)

The Kelvin berν(z) and beiν(z) functions satisfy the following differential equation: z2

d2 wðzÞ d wðzÞ +z − ði z2 + ν2 ÞwðzÞ = 0 2 dz dz

(2:1:17)

where wðzÞ = A Jν ði1=2 zÞ + B Kν ði1=2 zÞ

(2:1:18)

They can be defined in terms of the Bessel functions as Jν ðz i3=2 Þ = Jν ðz e3=4 π i Þ = berν ðzÞ + i beiν ðzÞ

(2:1:19)

Jν ðz i − 3=2 Þ = Jν ðz e − 3=4 π i Þ = berν ðzÞ− i beiν ðzÞ

(2:1:20)

and     

  1 1+i −1 + i −1 + i Jν − pffiffiffi z + Jν pffiffiffi z = Re Jν pffiffiffi z 2 2 2 2     

  1 1+i −1 + i −1 + i beiν ðzÞ = Jν − pffiffiffi z − i Jν pffiffiffi z = Im Jν pffiffiffi z 2 2 2 2 berν ðzÞ =

(2:1:21) (2:1:22)

where Re{ . . . } denotes the real part and Im{ . . . } the imaginary part of the Bessel functions. kerν(z) – Kelvin modified function, the real part of the Bessel function Kν(z) of order ν and of the argument i1/2z keiν(z) – Kelvin modified function, the imaginary part of the Bessel function Kν(z) of order ν and of the argument i–1/2z

8

2 Properties of the Bessel and Related Functions

ker0 ðzÞ = kerðzÞ

; kei0 ðzÞ = keiðzÞ

(2:1:23)

i − ν Kν ðz i 1=2 Þ = e− π ν i=2 Kν ðz eπ i=4 Þ = kerν ðzÞ + i keiν ðzÞ

(2:1:24)

i ν Kν ðz i − 1=2 Þ = e π ν i=2 Kν ðz e − π i=4 Þ = kerν ðzÞ− ikeiν ðzÞ

(2:1:25)

and

h  π ν i 1 + i  πν kerν ðzÞ = Re cos − i sin Kν pffiffiffi z 2 2 2

h     i πν πν 1+i keiν ðzÞ = Im cos − i sin Kν pffiffiffi z 2 2 2

(2:1:26) (2:1:27)

The Kelvin functions herν(z) and heiν(z) are kerν(z) and keiν(z) functions multiplied by factor ± 2/π her0 ðzÞ = herðzÞ herν ðzÞ =

;

hei0 ðzÞ = heiðzÞ

2 keiν ðzÞ π

2 heiν ðzÞ = − kerν ðzÞ π

(2:1:28) (2:1:29) (2:1:30)

Ai(z) – Airy function is expressed in terms of the modified Bessel function Kν(z) of order ν = 1/3 and the argument 2/3z3/2. Bi(z) – Airy function is expressed in terms of the modified Bessel functions Iν(z) of order ν = ± 1/3 and the argument 2/3z3/2. The Airy functions are particular solutions of the following differential equation: d2 wðzÞ − z wðzÞ = 0 dz2

(2:1:31)

wðzÞ = A AiðzÞ + B BiðzÞ

(2:1:32)

1 1 Aið0Þ = pffiffiffi Bið0Þ = 2=3 2 3 Γ 3 3

(2:1:33)

1 1 Ai′ð0Þ = − pffiffiffi Bi′ð0Þ = − 1=3  1 3 Γ 3 3

(2:1:34)

and

The Airy functions Ai(z), Ai(–z), Bi(z) and Bi(–z), with z > 0, are usually presented in the form

2.1 Definitions and Notations of the Bessel and Related Functions

rffiffiffiffiffiffiffiffi pffiffiffi  z z AiðzÞ = K1=3 ðζ Þ I−1=3 ðζ Þ− I1=3 ðζ Þ = 3π2 3

2 ; ζ = z3=2 3

 z z I−2=3 ðζ Þ− I2=3 ðζ Þ = − pffiffiffiffiffiffiffiffi K1=3 ðζ Þ 3 3π2 pffiffiffi  z Aið −zÞ = − J−1=3 ðζ Þ + J1=3 ðζ Þ 3   z Ai′ð −zÞ = J2=3 ðζ Þ− J−2=3 ðζ Þ 3 rffiffiffi  z BiðzÞ = I−1=3 ðζ Þ + I1=3 ðζ Þ 3 rffiffiffi  z Bið −zÞ = J−1=3 ðζ Þ− J1=3 ðζ Þ 3 Ai′ðzÞ = −

 z  Bi′ð−zÞ = pffiffiffi J−2=3 ðζ Þ + J2=3 ðζ Þ 3

9

(2:1:35) (2:1:36)

(2:1:37) (2:1:38) (2:1:39)

(2:1:40) (2:1:41)

or in terms of infinite integrals AiðzÞ =

1 π

Bið−zÞ =

 t3 + z t dt 3

(2:1:42)

3  t − z t dt 3

(2:1:43)

cos 0

1 Aið−zÞ = π 1 BiðzÞ = π



∞ ð

∞ ð

cos

0 ∞ ð

e

− t3 =3 + z t

3  t + sin + z t dt 3

(2:1:44)

0

1 π

∞ ð

e−t

3 =3 − z t

 + sin

t3 − zt 3

 dt

(2:1:45)

0

Gi(z) – Scorer function is expressed in terms of the Airy functions. The Scorer functions Gi(z) are particular solutions of the following differential equation d2 wðzÞ 1 − z wðzÞ = − dz2 π

(2:1:46)

wðzÞ = A AiðzÞ + B BiðzÞ + GiðzÞ

(2:1:47)

and

10

2 Properties of the Bessel and Related Functions

1 1 1 Gið0Þ = Bið0Þ = pffiffiffi Aið0Þ = 7=6 2 3 3 Γð3Þ 3

(2:1:48)

1 1 Gi′ð0Þ = − pffiffiffi Ai′ð0Þ = 5=6 1 3 Γð3Þ 3

(2:1:49)

Hi(z) – Scorer function is expressed in terms of Airy functions. The Scorer functions Hi(z) are particular solutions of the following differential equation d2 wðzÞ 1 − z wðzÞ = dz2 π

(2:1:50)

wðzÞ = A AiðzÞ + B BiðzÞ + HiðzÞ

(2:1:51)

2 2 2 Hið0Þ = Bið0Þ = − pffiffiffi Aið0Þ = 7=6 2 3 3 Γ 3 3

(2:1:52)

2 2 2 Hi′ð0Þ = Bi′ð0Þ = − Ai′ð0Þ = 5=6 1 3 3 3 Γ 3

(2:1:53)

and

The Scorer functions are expressed in terms of infinite integrals in the following way GiðzÞ =

1 π

GiðzÞ = 3

∞ ð

 3  t sin + z t dt 3

(2:1:54)

0

− 3=2

 2 ∞ ð K1=3 ðtÞ 2z dt π ζ 2 − t2

;

2 ζ = z3=2 3

(2:1:55)

0

Gið−zÞ = Bið−zÞ− 3

−3=2

 2 ∞ ð K1=3 ðtÞ 2z dt π ζ 2 + t2

(2:1:56)

0

pffiffiffi   2 z 2 3=2 S0, 1=3 z Gið−zÞ = Bið−zÞ− 3π 3

(2:1:57)

HiðzÞ = BiðzÞ− GiðzÞ

(2:1:58)

HiðzÞ =

1 π

∞ ð

e−t

3 =3 + z t

dt

(2:1:59)

 2 ∞ ð K1=3 ðtÞ 2z dt π ζ 2 + t2

(2:1:60)

0

Hið− zÞ = 3

−3=2

0

2.1 Definitions and Notations of the Bessel and Related Functions

Hi′ðzÞ =

∞ ð

1 π

te−t

3 =3 + z t

dt

11

(2:1:61)

0

Fν(z) and Gν(z) are the Bessel–Clifford functions of order ν defined by pffiffiffi Fν ðzÞ = z − ν=2 Cν ð2 zÞ ; Cν ðzÞ = Jν ðzÞ, Yν ðzÞ pffiffiffi Gν ðzÞ = z − ν=2 Zν ð2 zÞ ; Zν ðzÞ = Iν ðzÞ, Kν ðzÞ

(2:1:62) (2:1:63)

These functions satisfy the following differential equations z2

d2 wðzÞ d wðzÞ + ðν + 1Þz ± z wðzÞ = 0 d z2 dz

(2:1:64)

Solutions of differential equations when expressed in terms of the Bessel–Clifford functions can be interrelated by wðzÞ = zα Fν ðAzβ Þ =

zα ðAzβ Þ

ν=2

Cν ð2

wðzÞ = hðzÞ Fν ½gðzÞ = hðzÞ gðzÞ

pffiffiffiffiffiffiffiffi Azβ Þ

− ν=2

pffiffiffiffiffiffiffiffiffi Fν ½2 gðzÞ

(2:1:65)

Jiν(z), Yiν(z), liν(z) and Kiν(z) are the integral Bessel functions of order ν defined by ∞ ð

Jν ðtÞ dt t

Jiν ðzÞ = −

z ∞ ð

Yiν ðzÞ = −

Yν ðtÞ dt t

(2:1:62)

(2:1:63)

z

ðz Iiν ðzÞ = −

Iν ðtÞ dt t

0 ∞ ð

Kiν ðzÞ = −

Kν ðtÞ dt t

(2:1:64)

(2:1:65)

z

Frequently in the literature these functions are defined when minus sign is omitted. The integral defining Iiν(t) converges only for restricted range of orders. This function is rarely used, an example worth mentioning is when the integrand is replaced by (I0(t) – 1)/t [9]. Jin, k ðzÞand JiðkÞ n ðzÞ are the k–repeated integrals of the Bessel functions Jn(z) and Jn(z)/z, and Kin, k ðzÞand KiðkÞ n ðzÞ are the k–repeated integrals of the modified Bessel functions Kn(z) and Kn(z)/z, and are defined as follows:

12

2 Properties of the Bessel and Related Functions

ðz

ðz dt . . . Jn ðtÞdt

Jin, k ðzÞ =

JiðkÞ n ðzÞ =

0

0

ðz

ðz dt . . .

0

n≥0

(2:1:66)

; n>0

(2:1:67)

0

ðz

ðz

Kin, k ðzÞ =

KiðkÞ n ðzÞ =

Jn ðtÞ dt t

;

dt . . . Kn ðtÞdt 0

0

ðz

ðz dt . . .

0

Kn ðtÞ dt t

; n≥0

(2:1:68)

;

(2:1:69)

n>0

0

Kik(z) are the k–repeated integrals (also called the Bickley functions) defined by Ki0 ðzÞ = K0 ðzÞ

(2:1:70)

∞ ð

Kik ðzÞ =

Kik − 1 ðtÞ dt

; z>0 ;

k≥1

(2:1:71)

z

2.2 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Bessel Jν(z) and Yν(z) and the Hankel Hν(1)(z) and Hν(2)(z) functions h π ν π ν i Jν ði zÞ = iν Iν ðzÞ = cos + i sin Iν ðzÞ 2 2

(2:2:1)

Jν ð− zÞ = ð− 1Þν Jν ðzÞ = ½cosðπ νÞ + i sinðπ νÞJν ðzÞ

(2:2:2)

J−ν ðzÞ = Jν ðzÞ cosðπ νÞ− Yν ðzÞ sinðπ νÞ

(2:2:3)

Yn ðzÞ = lim ½cotðπ νÞ Jν ðzÞ− cscðπ νÞ J−ν ðzÞ ν!n

Y−ν ðzÞ = Jν ðzÞ sinðπ νÞ + Yν ðzÞ cosðπ νÞ π ν Yν ðzÞ + Y−ν ðzÞ = cot ½Jν ðzÞ− J−ν ðzÞ 2 π ν Yν ðzÞ− Y−ν ðzÞ = − tan ½Jν ðzÞ + J−ν ðzÞ 2

;

n = 0, ±1, ±2, ±3 . . .

(2:2:4) (2:2:5) (2:2:6) (2:2:7)

ð1Þ ðzÞ = eπ ν i Hνð1Þ ðzÞ H−ν

(2:2:8)

ð2Þ H−ν ðzÞ = e − π ν i Hνð2Þ ðzÞ

(2:2:9)

2.2 Interrelations, Recurrence Relations, Differential and Integral Formulas

13

Cν ðzÞ = Jν ðzÞ, Yν ðzÞ, Hνð1Þ ðzÞ, Hνð2Þ ðzÞ C−n ðzÞ = ð−1Þn Cn ðzÞ

;

n = 1, 2, 3, . . .

(2:2:10)

n

Cn ðzÞ = ð−1Þ Cn ð−zÞ

(2:2:11)

C−n ðzÞ = Cn ð−zÞ

(2:2:12)

Cν − 1 ðzÞ + Cν + 1 ðzÞ =

2ν Cν ðzÞ z

(2:2:13)

d ½C0 ðzÞ = −C1 ðzÞ dz

(2:2:14)

d ½z C1 ðzÞ = z C0 ðzÞ dz

(2:2:15)

d Cν ðzÞ Cν − 1 ðzÞ− Cν + 1 ðzÞ = dz 2

(2:2:16)

d Cν ðzÞ ν = Cν ðzÞ− Cν + 1 ðzÞ dz z

(2:2:17)

d Cν ðzÞ ν = Cν − 1 ðzÞ− Cν ðzÞ dz z

(2:2:18)

d ν ½z Cν ðzÞ = zν Cν − 1 ðzÞ dz

(2:2:19)

; ν≥0

d ν ½z Cν ðazÞ = a zν Cν − 1 ðazÞ dz d −ν ½z Cν ðzÞ = − z − ν Cν + 1 ðzÞ dz

(2:2:20) ;

ν≥0

d −ν ½z Cν ðazÞ = − a z − ν Cν + 1 ðazÞ dz   d k ν ½z Cν ðzÞ = zν − k Cν − k ðzÞ ; z dz 

d z dz

k

(2:2:21) (2:2:22)

k = 1, 2, 3, . . .

½z − ν Cν ðzÞ = ð− 1Þk z − ν − k Cν + k ðzÞ

;

k = 1, 2, 3, . . .

(2:2:23)

(2:2:24)

ð C1 ðzÞd z = − C0 ðzÞ

(2:2:25)

z C0 ðzÞ dz = z C1 ðzÞ

(2:2:26)

ð1 + z2 Þ C0 ðzÞdz = z C0 ðzÞ + z2 C1 ðzÞ

(2:2:27)

ð ð

14

2 Properties of the Bessel and Related Functions

ð ð

zν Cν − 1 ðzÞd z = zν Cν ðzÞ

(2:2:28)

  pffiffiffi 1 z ½Cν ðzÞH ′ν ðzÞ−C′ν ðzÞHν ðzÞ zν Cν ðzÞdz = 2ν − 1 π Γ ν + 2

1 ν≠− 2 ð z − ν Cν + 1 ðzÞd z = − z − ν Cν ðzÞ

(2:2:29)

(2:2:30)

ð ð ð ð ðz

z ln z C0 ðzÞdz = C0 ðzÞ + z ln z C1 ðzÞ

(2:2:31)

½2nzn − 1 + zn + 1 ln z Cn ðzÞ dz = zn Cn ðzÞ + z n + 1 ln z Cn + 1 ðzÞ

(2:2:32)

  pffiffiffi 1 z ½Cν ðzÞH ν − 1 ðzÞ− Cν −1 ðzÞH ν ðzÞ zν Cν ðzÞd z = 2ν − 1 π Γ ν + 2

(2:2:33)

z λ Cν ðzÞd z = ðλ + ν − 1Þ z Cν ðzÞ Sλ − 1, ν − 1 ðzÞ− z Cν −1 ðzÞ Sλ , ν ðzÞ

(2:2:34)

t − v Jν + 1 ðtÞdt =

0

ðz

tν Yν −1 ðtÞdt =

0

1 − z − ν Jν ðzÞ 2ν Γðν + 1Þ

2ν ΓðνÞ + z ν Yν ðzÞ π

½Jν ðzÞ2 −Jν − 1 ðzÞJν + 1 ðzÞ >

;

1 ½Jν ðzÞ2 ν+1

(2:2:35)

Re ν > 0 ; ν>0

(2:2:36)

;

z − real

(2:2:37)

ðz ln t J1 ðtÞ dt = ln 2 − γ − ln z J0 ðzÞ + Ji0 ðzÞ

(2:2:38)

0

2.3 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Modified Bessel Iν(z) and Kν(z) Functions In ð− zÞ = ð− 1Þn In ðzÞ I−n ðzÞ = In ðzÞ

;

;

n = 0, ± 1, ± 2, ± 3 . . .

n = 1, 2, 3, . . .

Iν ð− zÞ = ½cosðπ νÞ + i sinðπ νÞIν ðzÞ I−ν ðzÞ = Iν ðzÞ +

2 sinðπ νÞ Kν ðzÞ π

(2:3:1) (2:3:2) (2:3:3) (2:3:4)

2.3 Interrelations, Recurrence Relations, Differential and Integral Formulas

15

h π ν π ν i Iν ði zÞ = iν Jν ðzÞ = cos + i sin Jν ðzÞ 2 2

(2:3:5)

2ν Iν ðzÞ z

(2:3:6)

Iν − 1 ðzÞ− Iν + 1 ðzÞ = d ½I0 ðzÞ = I1 ðzÞ dz

(2:3:7)

d Iν ðzÞ Iν − 1 ðzÞ + Iν + 1 ðzÞ = dz 2

(2:3:8)

d Iν ðzÞ ν = Iν − 1 ðzÞ− Iν ðzÞ dz z

(2:3:9)

d Iν ðzÞ ν = Iν + 1 ðzÞ + Iν ðzÞ dz z  k d ½zν Iν ðzÞ = zν − k Iν − k ðzÞ z dz  ð ð ðz

d z dz

k

ð

;

k = 1, 2, 3, . . .

½z − ν Iν ðzÞ = z − ν − k Iν + k ðzÞ

(2:3:11)

(2:3:12)

zν + 1 Iν ðzÞdz = zν + 1 Iν + 1 ðzÞ

(2:3:13)

z − ν Iν + 1 ðzÞdz = z − ν Iν ðzÞ

(2:3:14)

t − ν Iν + 1 ðtÞd t = z − ν Iν ðzÞ−

0

ð

(2:3:10)

1 2ν Γðν + 1Þ

(2:3:15)

z1 − ν Iν ðzÞdz = z1 − ν Iν − 1 ðzÞ

(2:3:16)

zν Iν − 1 ðzÞdz = zν Iν ðzÞ

(2:3:17)

; Re ν > 0

  pffiffiffi 1 z ½Iν ðzÞLν − 1 ðzÞ− Iν −1 ðzÞLν ðzÞ zν Iν ðzÞd z = 2ν − 1 π Γ ν + 2   π I−ν ðzÞ− Iν ðzÞ ; ν ≠ 0, ± 1, ± 2, ± 3, . . . Kν ðzÞ = 2 sinðπ νÞ " # ∂ ðI−ν ðzÞ− Iν ðzÞÞ π ∂ν Kn ðzÞ = ; n = 0, 1, 2, 3, . . . ∂ 2 ∂ν ðsinðπ νÞÞ

ð

(2:3:18) (2:3:19)

(2:3:20)

ν=n

Kν ðzÞ = K−ν ðzÞ

(2:3:21)

16

2 Properties of the Bessel and Related Functions

Kν − 1 ðzÞ− Kν + 1 ðzÞ = −

2ν Kν ðzÞ z

(2:3:22)

d ½K0 ðzÞ = −K1 ðzÞ dz

(2:3:23)

d Kν ðzÞ Kν − 1 ðzÞ + Kν + 1 ðzÞ =− dz 2

(2:3:24)

dKν ðzÞ + ν Kν ðzÞ = − z Kν − 1 ðzÞ dz

z

(2:3:25)

dKν ðzÞ − ν Kν ðzÞ = − z Kν + 1 ðzÞ dz   d k ν ½z Kν ðzÞ = ð− 1Þk zν − k Kν − k ðzÞ z dz z



d z dz

ð ð ðz 0 ∞ ð

z

ð

k

(2:3:26) ; k = 1, 2, 3, . . .

½z − ν Kν ðzÞ = ð− 1Þk z − ν − k Kν + k ðzÞ

(2:3:27)

(2:3:28)

zν + 1 Kν ðzÞ d z = − zν + 1 Kν + 1 ðzÞ

(2:3:29)

z1 − ν Kν ðzÞdz = −z1 − ν Kν − 1 ðzÞ

(2:3:30)

tν Kν − 1 ðtÞd t = 2ν − 1 ΓðνÞ−z ν Kν ðzÞ

; Re ν > 0

(2:3:31)

t − ν Kν + 1 ðtÞd t = z − ν Kν ðzÞ

(2:3:32)

  pffiffiffi 1 z ½Kν ðzÞLν − 1 ðzÞ + Kν − 1 ðzÞLν ðzÞ zν Kν ðzÞdz = 2ν − 1 π Γ ν + 2

(2:3:33)

2.4 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Struve Hν(z) and the Modified Struve Lν(z) Functions H n ð− zÞ = ð− 1Þn + 1 H n ðzÞ H −1 ðzÞ =

2 − H 1 ðzÞ π

; n = 0, ± 1, ± 2, ± 3, . . .

(2:4:1) (2:4:2)

2.4 Interrelations, Recurrence Relations, Differential and Integral Formulas

H −2 ðzÞ = H 2 ðzÞ−

2 2z − π z 3π

H ν − 1 ðzÞ + H ν + 1 ðzÞ =

17

(2:4:3)

z ν 2ν H ν ðzÞ + pffiffiffi 2 z π Γ ν + 32

(2:4:4)

21 − ν sν, ν ðzÞ H ν ðzÞ = pffiffiffi  π Γ ν + 21

(2:4:5)

21 − ν Sν, ν ðzÞ H ν ðzÞ = Yν ðzÞ + pffiffiffi  π Γ ν + 21

(2:4:6)

d H 0 ðzÞ 2 = −H 1 ðzÞ dz π

(2:4:7)

d ½z H 1 ðzÞ = z H 0 ðzÞ dz

(2:4:8)

d H ν ðzÞ ν = H ν − 1 ðzÞ− H ν ðzÞ dz z

(2:4:9)

z ν d H ν ðzÞ ν = H ν ðzÞ− H ν + 1 ðzÞ + pffiffiffi 2 dz z 2 π Γ ν + 32 z ν d H ν ðzÞ H ν − 1 ðzÞ− H ν + 1 ðzÞ = + pffiffiffi 2 dz 2 2 π Γ ν + 32

(2:4:10)

(2:4:11)

d ν ½z H ν ðzÞ = zν H ν − 1 ðzÞ dz

(2:4:12)

d −ν 1 − z − ν H ν + 1 ðzÞ ½z H ν ðzÞ = ν pffiffiffi  dz 2 π Γ ν + 32

(2:4:13)

ðz

t1 + ν H ν ðtÞ dt = z1 + ν H ν + 1 ðzÞ

;

ν>−

1 2

(2:4:14)

0

ðz

t1 − ν H ν ðtÞ dt =

2ν − 1

0

ðz

t − ν H ν ðtÞdt =

2ν

0

z pffiffiffi  1 − z1 − ν H ν − 1 ðzÞ π Γ ν+2

z − z − ν H ν ðzÞ pffiffiffi  π Γ ν + 32

Lν ðzÞ = − ie − πνi H ν ði zÞ L1 ðzÞ =

2 − H 1 ðzÞ π

(2:4:15)

(2:4:16) (2:4:17) (2:4:18)

18

2 Properties of the Bessel and Related Functions

L2 ðzÞ =

2z −H 2 ðzÞ 3π

(2:4:19)

2 +L1 ðzÞ π

(2:4:20)

L−1 ðzÞ =

L−2 ðzÞ = L2 ðzÞ−

2 2z + πz 3π

(2:4:21)

z ν 2ν Lν ðzÞ + pffiffiffi 2 Lν − 1 ðzÞ− Lν + 1 ðzÞ = z π Γ ν + 32

(2:4:22)

d L0 ðzÞ 2 = L1 ðzÞ + dz π

(2:4:23)

z ν d Lν ðzÞ Lν − 1 ðzÞ + Lν + 1 ðzÞ = + pffiffiffi 2 dz 2 2 π Γ ν + 32

(2:4:24)

d ν ½z Lν ðzÞ = zν Lν − 1 ðzÞ dz

(2:4:25)

d −ν 1 + z − ν Lν + 1 ðzÞ ½z Lν ðzÞ = ν pffiffiffi  dz 2 π Γ ν + 32 ð 2z z ln z L0 ðzÞ dz = z ln z L1 ðzÞ− L0 ðzÞ + π ðz 2z L1 ðzÞ dt = L0 ðzÞ− π

(2:4:26)

(2:4:27) (2:4:28)

0

2.5 Interrelations, Recurrence Relations and Differential Formulas for the Lommel sµ,ν(z) and Sµ,ν(z) Functions sμ, − ν ðzÞ = sμ, ν ðzÞ

(2:5:1)

sμ + 2, ν ðzÞ = zμ + 1 − ½ðμ + 1Þ2 − ν2  sμ, ν ðzÞ

(2:5:2)

z ½ðμ + ν − 1Þ sμ − 1, ν − 1 ðzÞ− ðμ − ν − 1Þ sμ − 1, ν + 1 ðzÞ 2ν   pffiffiffi 1 H ν ðzÞ sν, ν ðzÞ = 2ν − 1 π Γ ν + 2

sμ, ν ðzÞ =

s−1, ν ðzÞ = − s0, ν ðzÞ =

π cscðπνÞ½J ν ðzÞ + J −ν ðzÞ 2ν

π cscðπ νÞ½J ν ðzÞ− J −ν ðzÞ 2

(2:5:3) (2:5:4) (2:5:5) (2:5:6)

2.5 Interrelations, Recurrence Relations and Differential Formulas

19

 ν d sμ, ν ðzÞ sμ, ν ðzÞ = ðμ − ν − 1Þ sμ − 1, ν − 1 ðzÞ− z dz

(2:5:7)

d sμ, ν ðzÞ ðμ + ν − 1Þ ðμ − ν − 1Þ sμ − 1, ν − 1 ðzÞ + sμ − 1, ν + 1 ðzÞ = 2 2 dz

(2:5:8)

 d  ν z sμ, ν ðzÞ = ðμ + ν − 1Þ sμ − 1, ν − 1 ðzÞ dz     μ−ν + 1 μ+ν+1 Γ Sλ, ν ðzÞ = sμ, ν ðzÞ + 2μ − 1 Γ 2 2 # "     π ðμ − νÞ π ðμ − νÞ Jν ðzÞ − cos Yν ðzÞ sin 2 2 Sμ, − ν ðzÞ = Sμ, ν ðzÞ

  pffiffiffi 1 Sν, ν ðzÞ = 2ν − 1 π Γ ν + ½H ν ðzÞ− Yν ðzÞ 2   pffiffiffi 1 Yν ðzÞ Sν, ν ðzÞ = sν, ν ðzÞ− 2ν − 1 π Γ ν + 2

(2:5:9)

(2:5:10)

(2:5:11) (2:5:12) (2:5:13)

S0, ν ðzÞ =

π cscðπ νÞ½J ν ðzÞ− J −ν ðzÞ− Jν ðzÞ + J−ν ðzÞ 2

(2:5:14)

S−1, ν ðzÞ =

π cscðπ νÞ½Jν ðzÞ + J−ν ðzÞ− J ν ðzÞ− J −ν ðzÞ 2ν

(2:5:15)

S1, ν ðzÞ = 1 + ν2 S−1, ν ðzÞ

(2:5:16)

Sμ + 2, ν ðzÞ = zμ + 1 − ½ðμ + 1Þ2 − ν2  Sμ, ν ðzÞ

(2:5:17)

Sμ, ν ðzÞ =

z ½ðμ + ν − 1Þ Sμ − 1, ν − 1 ðzÞ− ðμ − ν − 1Þ Sμ − 1, ν + 1 ðzÞ 2ν

S0, −1 ðzÞ =

1 z

(2:5:18) (2:5:19)

ν d Sμ, ν ðzÞ = ðμ − ν − 1Þ sμ − 1, ν − 1 ðzÞ− Sμ, ν ðzÞ dz z

(2:5:20)

ν d Sμ, ν ðzÞ Sμ, ν ðzÞ = ðμ − ν − 1Þ Sμ − 1, ν + 1 ðzÞ + z dz

(2:5:21)

d Sμ, ν ðzÞ ðμ + ν − 1Þ ðμ − ν − 1Þ Sμ − 1, ν − 1 ðzÞ + Sμ − 1, ν + 1 ðzÞ = 2 2 dz

(2:5:22)

20

2 Properties of the Bessel and Related Functions

2.6 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Anger Jν(z) and the Weber Eν(z) Functions J n ðzÞ = Jn ðzÞ

; n = 0, ± 1, ± 2, . . .

J ν − 1 ðzÞ+ J ν + 1 ðzÞ−

2ν 2 sinðπ νÞ J ν ðzÞ = − z πz

(2:6:1) (2:6:2)

J ν ðzÞ = cotðπ νÞ Eν ðzÞ− cscðπ νÞE−ν ðzÞ

(2:6:3)

J −ν ðzÞ = cosðπ νÞ J ν ðzÞ+ sinðπ νÞE ν ðzÞ

(2:6:4)

1 J ν ðzÞ = sinðπ νÞ ½s0, ν ðzÞ− νs−1, ν ðzÞ π

(2:6:5)

d J ν ðzÞ J ν − 1 ðzÞ− J ν + 1 ðzÞ = dz 2  d J ν ðzÞ ν sinðπ νÞ = J ν ðzÞ− J ν + 1 ðzÞ− dz z πz   ðz pffiffiffi 1 z ½J ν ðzÞLν − 1 ðzÞ−J ν − 1 ðzÞLν ðzÞ tν J ν ðtÞ dt = 2ν − 1 π Γ ν + 2 0

1 Re ν > − 2 ðz πz J 0 ðtÞ dt = z J 0 ðzÞ + ½J 0 ðzÞL1 ðzÞ− J 1 ðzÞL0 ðzÞ 2

(2:6:6) (2:6:7)

(2:6:8)

(2:6:9)

0

E−2n ðzÞ = E2n ðzÞ E2n ðzÞ = n

n X

ð− 1Þk

k=0

(2:6:10) z k − n Γð2n − kÞ H n − k ðzÞ Γðk + 1ÞΓðn − k + 1Þ 2

(2:6:11)

n = 1, 2, 3, . . . Eν − 1 ðzÞ + Eν + 1 ðzÞ −

2ν 2 ½cosðπ νÞ−1 Eν ðzÞ = z πz

(2:6:12)

Eν ðzÞ = cscðπ νÞ J −ν ðzÞ− cotðπ νÞ J ν ðzÞ

(2:6:13)

E−ν ðzÞ = cosðπ νÞ Eν ðzÞ− sinðπ νÞ J ν ðzÞ π ν 2 Eν ðzÞ = J ν ðzÞ tan − s0, ν ðzÞ 2 π

(2:6:14)

Eν ðzÞ = −

1 + cosðπ νÞ ν ½1 − cosðπ νÞ s0, ν ðzÞ− s−1, ν ðzÞ π π

(2:6:15) (2:6:16)

2.7 Interrelations, Recurrence Relations, Differential and Integral Formulas

d Eν ðzÞ Eν − 1 ðzÞ− Eν + 1 ðzÞ = dz 2  d Eν ðzÞ ν ½1 − cosðπνÞ = Eν ðzÞ− Eν + 1 ðzÞ− dz z πz

21

(2:6:17) (2:6:18)

2.7 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Kelvin berν(z), beiν(z), the Modified Kelvin kerν(z), keiν(z) and herν(z), heiν(z) Functions Rn ðzÞ = bern ðzÞ ; bein ðzÞ ; kern ðzÞ ; kein ðzÞ

;

n = 0, ± 1, ± 2, ...

R−n ðzÞ = ð− 1Þn R n ðzÞ bern ð− zÞ = ð− 1Þn bern ðzÞ ; n = 0, ± 1, ± 2, ± 3, . . .

pffiffiffi  2ν ½berν ðzÞ− beiν ðzÞ + berν − 1 ðzÞ berν + 1 ðzÞ = − z ber−ν ðzÞ = cosðπ νÞ berν ðzÞ + sinðπ νÞ beiν ðzÞ +

2 sinðπ νÞkerν ðzÞ π

ber−ν ðzÞ = cosðπ νÞ berν ðzÞ− sinðπ νÞ ½ heiν ðzÞ− beiν ðzÞ n

bein ð− zÞ = ð− 1Þ bein ðzÞ ; n = 0, ± 1, ± 2, ± 3, . . .

pffiffiffi  2ν ½beiν ðzÞ + berν ðzÞ + beiν − 1 ðzÞ beiν + 1 ðzÞ = − z bei−ν ðzÞ = − sinðπ νÞ berν ðzÞ + cosðπ νÞ beiν ðzÞ +

2 sinðπ νÞ keiν ðzÞ π

bei−ν ðzÞ = cosðπ νÞ beiν ðzÞ + sinðπ νÞ ½ herν ðzÞ− berν ðzÞ

(2:7:1) (2:7:2) (2:7:3) (2:7:4) (2:7:5) (2:7:6) (2:7:7) (2:7:8) (2:7:9)

d berðzÞ 1 = pffiffiffi ½ber1 ðzÞ + bei1 ðzÞ dz 2

(2:7:10)

d berν ðzÞ ν 1 = − berv ðzÞ− pffiffiffi ½berv − 1 ðzÞ + beiv − 1 ðzÞ dz z 2

(2:7:11)

d beiðzÞ 1 = pffiffiffi ½bei1 ðzÞ− ber1 ðzÞ dz 2

(2:7:12)

d beiν ðzÞ 1 = pffiffiffi ½ðbeiv + 1 ðzÞ− berv + 1 ðzÞÞ− ðbeiv − 1 ðzÞ− berv − 1 ðzÞÞ dz 8

(2:7:13)

d beiν ðzÞ ν 1 = beiv ðzÞ− pffiffiffi ½berv + 1 ðzÞ− beiv + 1 ðzÞ dz z 2

(2:7:14)

22

2 Properties of the Bessel and Related Functions

d beiν ðzÞ ν 1 = − berv ðzÞ + pffiffiffi ½berv − 1 ðzÞ−beiv − 1 ðzÞ dz z 2 ð 1 ber 1 ðzÞ dz = pffiffiffi ½berðzÞ− beiðzÞ− 1 2 ð zn zn bern − 1 ðzÞ dz = − pffiffiffi ½bern ðzÞ− bein ðzÞ 2 ð zn zn bein − 1 ðzÞdz = − pffiffiffi ½bern ðzÞ + bein ðzÞ 2

pffiffiffi  2ν kerν + 1 ðzÞ = − ½kerν ðzÞ− keiν ðzÞ + kerν − 1 ðzÞ z

(2:7:15) (2:7:16) (2:7:17) (2:7:18)

(2:7:19)

ker−ν ðzÞ = cosðπ νÞ kerν ðzÞ− sinðπ νÞ keiν ðzÞ

pffiffiffi  2ν ½keiν ðzÞ + kerν ðzÞ + keiν − 1 ðzÞ keiν + 1 ðzÞ = − z

(2:7:20)

kei−ν ðzÞ = sinðπ νÞ kerν ðzÞ + cosðπ νÞ keiν ðzÞ

(2:7:22)

d kerν ðzÞ 1 = pffiffiffi f½kerν + 1 ðzÞ + keiv + 1 ðzÞ − ½kerv − 1 ðzÞ + keiv − 1 ðzÞg dz 8

(2:7:23)

d keiν ðzÞ 1 = pffiffiffi f½keiv + 1 ðzÞ− kerv + 1 ðzÞ − ½keiv − 1 ðzÞ− kerv − 1 ðzÞg dz 8

(2:7:24)

hern ð− zÞ = ð− 1Þn ½hern ðzÞ− 2 bern ðzÞ

(2:7:25)

;

n = 0, ± 1, ± 2, ± 3, . . .

(2:7:21)

her−ν ðzÞ = cosðπ νÞ herν ðzÞ− sinðπ νÞ heiν ðzÞ

(2:7:26)

hein ð− zÞ = ð− 1Þn ½hein ðzÞ− 2 bein ðzÞ

(2:7:27)

; n = 0, ± 1, ± 2, ± 3, . . .

hei−ν ðzÞ = sinðπ νÞ herν ðzÞ + cosðπ νÞ heiν ðzÞ

(2:7:28)

2.8 Interrelations, Recurrence Relations, Differential and Integral Formulas for the Bessel Integral Functions Ji0 ð0Þ = cið0Þ − ln 2

(2:8:1) ðz

Ji0 ðzÞ = ciðzÞ − ln 2 +

J0 ðxÞ − cos x dx x

(2:8:2)

0

z ðz J ðxÞ − 1 0 + dx Ji0 ðzÞ = γ + ln 2 x 0

(2:8:3)

2.9 Series Expansions of the Bessel and Related Functions

Ji2 ðzÞ = −

J1 ðzÞ z

(2:8:4)

Ji2 n ðzÞ = Ji2 n ð− zÞ Ji2 n + 1 ðzÞ + Ji2 n Jin ðzÞ =

23

;

n = 0, 1, 2, 3, . . .

(2:8:5)

2 2n + 1

(2:8:6)

+ 1 ð− zÞ =

−

z ½ðn − 1ÞJin − 1 ðzÞ− ðn + 1ÞJin + 1 ðzÞ 2n

z ½ð2n − 2ÞJi2 n − 2 ðzÞ− 2n Ji2 n ðzÞ ð4n − 2Þ ð  1 Jiν ðzÞ = Jν − 1 ðzÞdz − Jν ðzÞ− 1 dz ν

Ji2 n − 1 ðzÞ =

(2:8:7) (2:8:8) (2:8:9)

d Ji0 ðzÞ J0 ðzÞ = dz z

(2:8:10)

d Yi0 ðzÞ Y0 ðzÞ = dz z

(2:8:11)

d Jin ðzÞ Jin ðzÞ = dz z

(2:8:12)

d Jin ðzÞ ½ðn − 1ÞJin − 1 ðzÞ−ðn + 1ÞJin + 1 ðzÞ = dz 2n ∞ ð pffiffiffiffiffi sin x ciðxÞ siðtÞ Ji0 ð2 x tÞ dt = − 2x 2 0 ∞ ð

0 ∞ ð

0 ∞ ð

0

(2:8:13) (2:8:14)

pffiffiffiffiffi 1 − cos x siðxÞ ciðtÞ Ji0 ð2 x tÞ dt = − 2x 2

(2:8:15)

pffiffiffiffiffi dt siðtÞ Ji1 ð2 x tÞ pffiffi = t

(2:8:16)

π 2

pffiffiffi + siðxÞ + 2 sin x pffiffiffi + 4Cð xÞ x

pffiffiffiffiffi dt γ + 2 + ln x − ciðxÞ− 2 cos x pffiffiffi pffiffiffi + 4Sð xÞ ciðtÞ Ji1 ð2 xtÞ pffiffi = x t

(2:8:17)

2.9 Series Expansions of the Bessel and Related Functions Jn ðzÞ =

∞ X ð− 1Þk + n k=0

z 2k + n 2

k!ðk + nÞ!

(2:9:1)

24

2 Properties of the Bessel and Related Functions

∞ X ð− 1Þk

J± ν ðzÞ =

k=0

z 2k ±

ν

2

(2:9:2)

k!Γðk ± ν + 1Þ

  k   n−1 2 ln z2 1 2 nX ðn − k − 1Þ! z2 + Jn ðzÞ 4 π π z k=0 k!  2 k ∞  − z4 X n 1 z − ½ψðk + 1Þ + ψðn + k + 1Þ π 2 k = 0 k!ðn + kÞ!

Yn ðzÞ = −

In ðzÞ =

z 2k +

∞ X

Iν ðzÞ =

(2:9:4)

k!ðk + nÞ! z 2 k +

∞ X k=0

n

2

k=0

ν

2

(2:9:5)

k!Γðk + ν + 1Þ

 k n−1 z 1 2n X ðn − k − 1Þ! z2 ð− 1Þk Kn ðzÞ = ð− 1Þn + 1 In ðzÞ ln + 4 2 2 z k=0 k!   k ∞ ð− 1Þn z n X ½ψðk + 1Þ + ψðn + k + 1Þ z2 + 2 4 2 k=0 k!ðn + kÞ! z 2k + ν + 1 H ν ðzÞ = ð− 1Þ  23  Γ k + 2 Γ k + ν + 32 k=0  n=2 1 X Γ k + 21 z n − 2k − 1  − H n ðzÞ Ln ðzÞ = π k = 0 Γ n + 21 − k 2 ∞ X

L−n ðzÞ =

(2:9:3)

k

(2:9:7)

; n = 0, 1, 2, 3, . . .

  n=2  ð− 1Þn + 1 X Γ n − k − 21 2 n − 2k − 1  − H −n ðzÞ π z Γ k + 32 k=0

z 2k + ν + 1  2  Lν ðzÞ = Γ k + 32 Γ k + ν + 32 k=0 ∞ X

(2:9:8)

(2:9:9)

(2:9:10)

 2k   ∞ ð− 1Þk z2 Γ μ − 2ν + 1 Γ μ + 2ν + 1 zμ + 1 X   sμ, ν ðzÞ = 4 k = 0 Γ μ − ν + 2k + 3 Γ μ + ν + 2k + 3 2

(2:9:6)

(2:9:11)

2

)  2k (   2 ∞ X ð− 1Þk z2 π2 z ψ′ðk + 1Þ π2 ln − ψðk + 1Þ − J0 ðzÞ + + S−1, 0 ðzÞ = 8 2 2 2 2 ðk!Þ2 k=1 (2:9:12)

2.9 Series Expansions of the Bessel and Related Functions

25

Sν − 1, ν ðzÞ = − 2ν − 2 π ΓðνÞ Yν ðzÞ +

 2k h  ∞ i ð− 1Þk z2 ΓðνÞ zν X z 2 ln − ψðk + ν + 1Þ− ψðk + 1Þ 4 k = 0 k!Γðk + ν + 1Þ 2

z 2k ∞  π ν X 2  ð− 1Þk  J ν ðzÞ = cos 2 k=0 Γ k + ν2 + 1 Γ k − ν2 + 1 z 2k + 1 ∞  π ν X 2  ð− 1Þk  + sin 2 k=0 Γ k + ν +2 3 Γ k + 3 −2 ν z 2k ∞ π ν X 2  ð− 1Þk  2 k=0 Γ k + ν2 + 1 Γ k − ν2 + 1 z 2k + 1 ∞  π ν X 2  ð− 1Þk  − cos 2 k=0 Γ k + ν +2 3 Γ k + 3 −2 ν h i  ð2k − 1Þ π k ∞ 3ðk − 2Þ=3 Γ k + 1 cos z X 3 3 1

(2:9:13)

(2:9:14)

Eν ðzÞ = sin

GiðzÞ =

π k=0

k!  ∞ ðk − 2Þ=3 k + 1 3 Γ 3 zk 1X HiðzÞ = k! π k=0

beiðzÞ =

(2:9:18)

   k ∞ z ν X cos π 3 ν + 4 k z2 4

2

k=0

k!Γðν + k + 1Þ

2

;

z≥0

 2 4 k + 2 z 2 2 ½ð2k + 1Þ! k=0

∞ X

beiν ðzÞ =

ð− 1Þk

4

k=0

k!Γðν + k + 1Þ

(2:9:19)

(2:9:20)

    k ∞ z ν X sin π 3 ν + 4 k z2 2

(2:9:16)

(2:9:17)

 4 k ∞ X ð− 1Þk z2 berðzÞ = 2 2 k = 0 ½ð2kÞ! berν ðzÞ =

(2:9:15)

2

;

z≥0

(2:9:21)

 2k ∞ z X π ψð2k + 1Þ z2 ð− 1Þk kerðzÞ = − ln berðzÞ + beiðzÞ + 2 4 ½ð2kÞ!2 4 k= 0

(2:9:22)

 2k ∞ z X π ψð2k + 2Þ z2 keiðzÞ = − ln ð− 1Þk beiðzÞ− berðzÞ + 2 4 ½ð2k + 1Þ!2 4 k= 0

(2:9:23)

Jin ðzÞ =

z n  ∞   X ð− 1Þk z n + 2k 1 2 + −1+ 2 n! ðn + 2kÞk!ðn + kÞ! n k=1

(2:9:24)

26

2 Properties of the Bessel and Related Functions

z ν  ∞  ν + 2k  X ð− 1Þk z2 1 2 + −1+ Jiν ðzÞ = Γðν + 1Þ ðν + 2kÞk!Γðν + k + 1Þ ν k=1  2k   X ∞ ð− 1Þk z2 z + Ji0 ðzÞ = γ + ln 2 2 k = 1 2n ðn!Þ

Yi0 ðzÞ =

(2:9:25)

(2:9:26)

γ2 π 2γ z 1 h z i2 − + ln + ln π 6 π 2 π 2 z 2k  k ∞ z  2 X ð− 1Þ 2 1 − ψðn + 1Þ + − ln π k = 1 2n ðn!Þ2 2n 2

(2:9:27)

z 1 h z i2 γ2 π 2 − − γ ln − ln 2 24 2 2 2   2k ∞ z  z 2X 1 2 + ψðn + 1Þ + − ln π k = 1 2n ðn!Þ2 2n 2

(2:9:28)

Ki0 ðzÞ = −

∞ h z i2 h z i2 X +2 ð− 1Þk Jk J0 ðzÞ = J0 2 2 k=1 ( )  ∞ z k z i X 2 h 2 Y0 ðzÞ = Jk ðzÞ γ + ln J0 ðzÞ + k!k π 2 k=0

( ) ∞ z i X 2 h ð− 1Þk Y0 ðzÞ = J2k ðzÞ γ + ln J0 ðzÞ− 2 k π 2 k=0

I0 ðzÞ = 1 − 2

∞ X

ð− 1Þk I2 k ðzÞ

(2:9:29)

(2:9:30)

(2:9:31)

(2:9:32)

k=1

I0 ðzÞ = e − z − 2 I0 ðzÞ = ez − 2

∞ X

ð− 1Þk Ik ðzÞ

k=1 ∞ X

Ik ðzÞ

(2:9:33) (2:9:34)

k=1

I1 ðzÞ =

∞ 2X k I2 k ðzÞ z k=1

z k + ν ∞ 1 X 2 Jν ðzÞ = Jk ðzÞ ΓðνÞ k = 0 k!ðk + νÞ Jν ðzÞ =

∞ X

ð− 1Þk

k=0

Iν ðzÞ =

∞ k X z k=0

k!

zk Ik + ν ðzÞ k!

Jk + ν ðzÞ

(2:9:35)

(2:9:36)

(2:9:37)

(2:9:38)

27

2.10 Bessel and Related Functions of Half Odd Integer Order

 k + ν ∞ ð− 1Þk z2 1 X Iν ðzÞ = Ik ðzÞ ΓðνÞ k = 0 k!ðk + νÞ

(2:9:39)

 k ∞ z i X ð− 1Þk z2 K0 ðzÞ = − γ + ln Ik ðzÞ I0 ðzÞ − k!k 2 k=0 h

∞ h z i X 1 K0 ðzÞ = − γ + ln I0 ðzÞ + 2 I2k ðzÞ 2 k k=0

sλ, ν ðzÞ = 2λ + 1

(2:9:40)

∞ X

ð2k + λ + 1Þ Γðk + λ + 1Þ J2k + λ + 1 ðzÞ k!ð2k + λ + ν + 1Þð2k + λ − ν + 1Þ k=1

2λ + 1 Γðλ + 2Þ + Jλ + 1 ðzÞ ðλ + ν + 1Þðλ − ν + 1Þ

;

λ ± ν ≠ n = −1, −3, −5, . . .

∞ X 4 ð2k + ν + 1Þ Γðk + ν + 1Þ H ν ðzÞ = pffiffiffi  J2k + ν + 1 ðzÞ 1 π Γ ν + 2 k = 0 k!ð2k + 2ν + 1Þð2k + 1Þ

H ν ðzÞ = H 0 ðzÞ =

z k rffiffiffiffiffiffi ∞ z X  2 1 Jk + ν + 1=2 ðzÞ 2 π k = 0 k! k + 2 ∞ X

(2:9:41)

ð−1Þk ½Jk + 1=2 ðzÞ2

(2:9:42)

(2:9:43) (2:9:44)

k=0

H 0 ðzÞ =

∞ 4X J2k + 1 ðzÞ π k = 0 2k + 1

(2:9:45)

H 1 ðzÞ =

∞ 2 4X J2k ðzÞ ½1 − J0 ðzÞ + π π k = 1 4k2 − 1

(2:9:46)

2.10 Bessel and Related Functions of Half Odd Integer Order rffiffiffiffiffiffi      2 15 6 15 1 − 2 sin z + J−7=2 ðzÞ = − 3 cos z πz z z z rffiffiffiffiffiffi     2 3 sin z 3 J−5=2 ðzÞ = + 2 − 1 cos z πz z z rffiffiffiffiffiffi h i 2 cos z J−3=2 ðzÞ = − + sin z πz z rffiffiffiffiffiffi 2 cos z J−1=2 ðzÞ = πz rffiffiffiffiffiffi 2 sin z J1=2 ðzÞ = πz

(2:10:1)

(2:10:2)

(2:10:3)

(2:10:4) (2:10:5)

28

2 Properties of the Bessel and Related Functions

rffiffiffiffiffiffi   2 sin z J3=2 ðzÞ = − cos z πz z ! # rffiffiffiffiffiffi " 2 3 3 − 1 sin z − cos z J5=2 ðzÞ = πz z2 z rffiffiffiffiffiffi      2 15 6 15 sin z − − − 1 cos z J7=2 ðzÞ = π z z3 z z2 rffiffiffiffiffiffi 2 Y−1=2 ðzÞ = sin z πz rffiffiffiffiffiffi 2 Y1=2 ðzÞ = − cos z πz rffiffiffiffiffiffi  2 cos z Y3=2 ðzÞ = − sin z + πz z rffiffiffiffiffiffi    2 3 3 1− cos z − sin z Y5=2 ðzÞ = πz z z J−n − 1=2 ðzÞ = ð− 1Þn + 1 Yn + 1=2 ðzÞ

;

n = 0, 1, 2, 3, . . .

Y−n − 1=2 ðzÞ = ð− 1Þn Jn + 1=2 ðzÞ ; n = 0, 1, 2, 3, . . . rffiffiffiffiffiffi      2 15 15 6 cosh z + 1 sinh z − + I−7=2 ðzÞ = π z z2 z3 z rffiffiffiffiffiffi    2 3 3 + 1 cosh z − I−5=2 ðzÞ = sinh z π z z2 z rffiffiffiffiffiffi   2 cosh z sinh z − I−3=2 ðzÞ = πz z rffiffiffiffiffiffi 2 I−1=2 ðzÞ = cosh z πz rffiffiffiffiffiffi 2 I 1=2 ðzÞ = sinh z πz rffiffiffiffiffiffi   2 sinh z I 3=2 ðzÞ = cosh z − πz z rffiffiffiffiffiffi    2 3 3 I5=2 ðzÞ = + 1 sinh z − cosh z π z z2 z

(2:10:6) (2:10:7)

(2:10:8)

(2:10:9)

(2:10:10)

(2:10:11)

(2:10:12) (2:10:13) (2:10:14) (2:10:15)

(2:10:16)

(2:10:17)

(2:10:18)

(2:10:19)

(2:10:20)

(2:10:21)

2.10 Bessel and Related Functions of Half Odd Integer Order

rffiffiffiffiffiffi      2 15 15 6 sinh z I7=2 ðzÞ = + 1 cosh z − − π z z2 z3 z rffiffiffiffiffi π −z K−1=2 ðzÞ = e 2z rffiffiffiffiffi π −z K 1=2 ðzÞ = e 2z rffiffiffiffiffi   π −z 1 K 3=2 ðzÞ = 1+ e 2z z rffiffiffiffiffi   π −z 3 3 K 5=2 ðzÞ = 1+ + 2 e 2z z z 2 I−n − 1=2 ðzÞ = ð− 1Þn Kn + 1=2 ðzÞ + In + 1=2 ðzÞ π

;

n = 0, 1, 2, 3, . . .

K−n − 1=2 ðzÞ = Kn + 1=2 ðzÞ ; n = 0, 1, 2, 3, . . . rffiffiffiffiffiffi  2 sin z cos z − H −3=2 ðzÞ = πz z rffiffiffiffiffiffi 2 H −1=2 ðzÞ = sin z πz rffiffiffiffiffiffi 2 H 1=2 ðzÞ = ð1 − cos zÞ πz rffiffiffiffiffiffi   rffiffiffiffiffiffi z 2 2 cos z H 3=2 ðzÞ = sin z + 1+ 2 − 2π z πz z 2 I−n − 1=2 ðzÞ = ð− 1Þn Kn + 1=2 ðzÞ + In + 1=2 ðzÞ ; n = 0, 1, 2, 3, . . . π rffiffiffiffiffiffi  2 sinh z cosh z − L−3=2 ðzÞ = πz z rffiffiffiffiffiffi 2 L−1=2 ðzÞ = sinh z πz rffiffiffiffiffiffi 2 L1=2 ðzÞ = ð1 − cosh zÞ πz rffiffiffiffiffiffi   rffiffiffiffiffiffi  z 2 2 cosh z L3=2 ðzÞ = − sinh z − 1− 2 + 2π z πz z L−n − 1=2 ðzÞ = In + 1=2 ðzÞ

; n = 0, 1, 2, 3, . . .

29

(2:10:22)

(2:10:23)

(2:10:24)

(2:10:25)

(2:10:26) (2:10:27) (2:10:28) (2:10:29)

(2:10:30)

(2:10:31)

(2:10:32) (2:10:33) (2:10:34)

(2:10:35)

(2:10:36)

(2:10:37) (2:10:38)

30

2 Properties of the Bessel and Related Functions

ber−1=2 ðzÞ rffiffiffiffiffiffi             2 3π z z 3π z z cos cos pffiffiffi cosh pffiffiffi + sin sin pffiffiffi sinh pffiffiffi = πz 8 8 2 2 2 2 (2:10:39) bei−1=2 ðzÞ rffiffiffiffiffiffi             2 3π z z 3π z z cos sin pffiffiffi sinh pffiffiffi − sin cos pffiffiffi cosh pffiffiffi = πz 8 8 2 2 2 2 (2:10:40)

    pffiffi pffiffi 1 z π z π ber1=2 ðzÞ = pffiffiffiffiffiffiffiffi ez= 2 cos pffiffiffi + − e − z= 2 cos pffiffiffi − 2πz 2 8 2 8

pffiffi     pffiffi 1 z π z π − e − z= 2 sin pffiffiffi − bei1=2 ðzÞ = pffiffiffiffiffiffiffiffi ez= 2 sin pffiffiffi + 2πz 2 8 2 8 rffiffiffiffiffi pffiffi   π z= 2 z π ker−1=2 ðzÞ = cos pffiffiffi − e 2z 2 8 rffiffiffiffiffi   π − z=pffiffi2 z π kei−1=2 ðzÞ = − sin pffiffiffi − e 2z 2 8 rffiffiffiffiffi pffiffi   π z= 2 z 3π ker1=2 ðzÞ = cos pffiffiffi + e 2z 8 2 rffiffiffiffiffi   π − z=pffiffi2 z 3π kei1=2 ðzÞ = − sin pffiffiffi + e 2z 8 2 rffiffiffiffiffiffi 2 J −1=2 ðzÞ = fcos z½CðzÞ + SðzÞ − sin z½CðzÞ− SðzÞg πz rffiffiffiffiffiffi 2 J 1=2 ðzÞ = fcos z½CðzÞ− SðzÞ + sin z½CðzÞ + SðzÞg πz rffiffiffiffiffiffi 2 E−1=2 ðzÞ = − fcos z½CðzÞ− SðzÞ + sin z½CðzÞ + SðzÞg πz rffiffiffiffiffiffi 2 E1=2 ðzÞ = fcos z½CðzÞ + SðzÞ − sin z½CðzÞ− SðzÞg πz rffiffiffiffiffiffi 2π s−1, 1=2 ðzÞ = 2 ½sin z SðzÞ + cos z CðzÞ z rffiffiffiffiffiffi 2π s0, 1=2 ðzÞ = ½sin z CðzÞ− cos z SðzÞ z

(2:10:41) (2:10:42)

(2:10:43)

(2:10:44)

(2:10:45)

(2:10:46)

(2:10:47)

(2:10:48)

(2:10:49)

(2:10:50)

(2:10:51)

(2:10:52)

2.11 Bessel and Related Functions Expressed in Terms of Special Functions

31

s1=2, 1=2 ðzÞ =

1 − cos z pffiffiffi z

(2:10:53)

s3=2, 1=2 ðzÞ =

z − sin z pffiffiffi z

(2:10:54)

z2 + 2 cos z − 2 pffiffiffi z 1 S−3=2, 1=2 ðzÞ = − pffiffiffi ½sin z siðzÞ + cos z CiðzÞ z rffiffiffiffiffiffi     2π 1 1 cos z − CðzÞ + sin z − SðzÞ S−1, 1=2 ðzÞ = 2 z 2 2

s5=2, 1=2 ðzÞ =

1 S−1=2, 1=2 ðzÞ = pffiffiffi ½sin z CiðzÞ− cos z siðzÞ z rffiffiffiffiffiffi     2π 1 1 S0, 1=2 ðzÞ = cos z − SðzÞ − sin z − CðzÞ z 2 2 pffiffiffi S3=2, 1=2 ðzÞ = z

(2:10:55) (2:10:56) (2:10:57) (2:10:58)

(2:10:59) (2:10:60)

1 S1=2, 1=2 ðzÞ = pffiffiffi z

(2:10:61)

  pffiffiffi 23=2 sin z p ffiffi ffi p ffiffiffi Ji1=2 ðzÞ = − + 2Cð z Þ z π

(2:10:62)

2.11 Bessel and Related Functions Expressed in Terms of Special Functions z ν Jν ðzÞ =

2

Γðν + 1Þ

0 F 1 ð;ν + 1;

−

z2 Þ 4

(2:11:1)

z ν

  1 ν + 1 ν + 2 z4 Jν ðzÞ = , ; 0F3 ; . Γðν + 1Þ 2 2 2 256 z ν + 2   3 ν + 2 ν + 3 z4 + 2 , ; 0F3 ; . Γðν + 2Þ 2 2 2 256 z ν   z2 2 Iν ðzÞ = 0 F 1 ;ν + 1; Γðν + 1Þ 4 2

(2:11:2)

(2:11:3)

z ν Iν ðzÞ =

  e−z 1 1 F 1 ν + ;1 + 2ν;2z Γðν + 1Þ 2 2

(2:11:4)

32

2 Properties of the Bessel and Related Functions

pffiffiffi pffiffiffi π K1=4 ðzÞ = 1=4 D−1=2 ð2 zÞ z   pffiffiffi 1 Kν ðzÞ = ð2 zÞν π e − z 1 F1 ν + 2 ν + 1; 2 z 2 rffiffiffiffiffi π W0 , ν ð2zÞ Kν ðzÞ = 2z  z ν + 1  3 2 3 z2 H ν ðzÞ = pffiffiffi  1 F2 1; , ν + ; − 4 2 2 π Γ ν + 32 2  z ν + 1  2 3 z2 Lν ðzÞ = pffiffiffi  F 1; ν + ; 1 2 2 4 π Γ ν + 32 2     1 2 z3 z 4 z3 − 1=3  1 0 F1 ; ; AiðzÞ = 2=3 2 0 F1 ; ; 3 9 3 9 3 Γ 3 3 Γ 3     1 2 z3 z 4 z3 Aið−zÞ = 2=3 2 0 F1 ; ; − + 1=3 1 0 F1 ; ; − 9 9 3 3 3 Γ 3 3 Γ 3     1 1 z3 z2 5 z3  1 0 F1 ; ;  2 0 F1 ; ; + 3 9 3 9 31=3 Γ 3 2 · 32=3 Γ 3     1 1 z3 z2 5 z3  2 0 F1 ; ; − Ai′ð− zÞ = − 1=3  1 0 F1 ; ; − + 9 9 3 3 3 Γ 3 2 · 32=3 Γ 3

Ai′ðzÞ = −

BiðzÞ =

    1 2 z3 31=6 z 4 z3  2 0 F1 ; ;  1 0 F1 ; ; + 3 9 3 9 31=6 Γ 3 Γ 3

Bið− zÞ =

  1=6   1 2 z3 3 z 4 z3  2 0 F1 ; ; −  1 0 F1 ; ; − − 9 9 3 3 31=6 Γ 3 Γ 3

    31=6 1 z3 z2 5 z3 ′   Bi ðzÞ = 1 0 F1 ; ; + 0 F1 ; ; 3 9 3 9 Γ 3 2 · 31=6 Γ 32     31=6 1 z3 z2 5 z3  2 0 F1 ; ; − Bi′ð− zÞ =  1 0 F1 ; ; − + 9 9 3 3 Γ 3 2 · 31=6 Γ 3 pffiffiffi   2 z 2 Gið− zÞ = Bið− zÞ− S0, 1=3 z3=2 3π 3   1 1 z4 berðzÞ = 0 F3 ; , , 1; − 256 2 2

(2:11:5) (2:11:6)

(2:11:7) (2:11:8)

(2:11:9)

(2:11:10)

(2:11:11)

(2:11:12)

(2:11:13)

(2:11:14)

(2:11:15)

(2:11:16)

(2:11:17)

(2:11:18)

(2:11:19)

2.12 Integral Representation of the Bessel and Related Functions

   cos 3 π4 ν 1 ν+1 ν+2 z4 F ; . , ; − 0 3 256 Γðν + 1Þ 2 2 2  z ν + 2 3 π ν   sin 4 3 ν+2 ν+3 z4 2 − , ;− 0 F3 ; . 256 Γðν + 2Þ 2 2 2   z2 3 3 z4 beiðzÞ = 0 F3 ; , , 1; − 4 256 2 2  z ν    sin 3 π4 ν 1 ν+1 ν+2 z4 beiν ðzÞ = 2 F ; . , ; − 0 3 256 Γðν + 1Þ 2 2 2 z ν + 2  3 π ν   cos 4 3 ν+2 ν+3 z4 2 + . , ;− 0F3 ; 256 Γðν + 2Þ 2 2 2   dberðzÞ z3 3 3 z4 =− 0 F3 ; 2, , ; − 256 dz 2 2 256   dbeiðzÞ z 1 3 z4 = 0 F3 ; 1, , ; − dz 2 2 2 256   zλ + 1 λ − ν + 3 λ + ν + 3 z2 sλ, ν ðzÞ = , ;− 1 F2 1; ðλ + ν + 1Þðλ − ν + 1Þ 4 2 2

33

 z ν

berν ðzÞ =

2

λ ± ν ≠ − 1, − 2, − 3, . . .   z z 2 z2 Ji0 ðzÞ = γ + ln − 2 F3 1, 1; 2, 2, 2; − 8 4 2

(2:11:20)

(2:11:21)

(2:11:22)

(2:11:23)

(2:11:24)

(2:11:25)

(2:11:26)

2.12 Integral Representation of the Bessel and Related Functions 2 J0 ðzÞ = π 1 J0 ðzÞ = π 2 J0 ðzÞ = π

∞ ð

1

ð1 −1

ð1 0

sinðz tÞ pffiffiffiffiffiffiffiffiffiffi dt t2 − 1

(2:12:1)

cosðz tÞ pffiffiffiffiffiffiffiffiffiffi dt 1 − t2

(2:12:2)

cosðz tÞ pffiffiffiffiffiffiffiffiffiffi dt 1 − t2

(2:12:3)

ðπ 1 cosðz cos tÞ dt J0 ðzÞ = π 0

34

2 Properties of the Bessel and Related Functions

J0 ðzÞ =

J0 ðzÞ =

2 π2 2 π

ðπ t cosðz sin tÞdt

(2:12:4)

0 ∞ ð

sinðz cosh tÞ dt

;

z>0

(2:12:5)

;

n = 0, 1, 2, 3, . . .

(2:12:6)

; n = 0, 1, 2, 3, . . .

(2:12:7)

0

ðπ

Jn ðzÞ =

1 cos zðsin t − n tÞ dt π 0

1 Jn ðzÞ = 2π

2ðπ

cos zðn t − sin tÞ dt 0

π=2  ν ð 2 z2 Jν ðzÞ = pffiffiffi  cosðz cos tÞðsin tÞ2 ν dt π Γ ν + 21

; Re ν > −

1 2

(2:12:8)

0

z ν ðπ 2 cosðz cos tÞðsin tÞ2 ν dt Jν ðzÞ = pffiffiffi π Γ ν + 21

;

1 2

Re ν > −

(2:12:9)

0

π=2  ν ð 2 z cosðz sin tÞðcos tÞ2 ν dt Jν ðzÞ = pffiffiffi 2 1 πΓ ν+ 2

; Re ν > −

1 2

(2:12:10)

0

z ν Jν ðzÞ = pffiffiffi 2 π Γ ν + 21 2 Jν ðzÞ = π

π=2 ð

cosðz sin tÞðcos tÞ2 ν dt

;

Re ν > −

1 2

(2:12:11)

−π=2

∞ ð

sinðz cosh t −

πν Þ coshðν tÞ dt 2

;

jRe νj < 1

(2:12:12)

0

 ν ð1 2 z2 ν − 1=2 ð1 − t2 Þ cosðz tÞdt Jν ðzÞ = pffiffiffi  π Γ ν + 21

;

Re ν > −

1 2

(2:12:13)

z ν ð1 ν − 1=2 pffiffiffiffiffiffiffiffi t 2 pffiffiffiffiffiffiffiffi cosðz 1 − tÞdt Jν ðzÞ = pffiffiffi 1 πΓ ν+ 2 1−t

;

Re ν > −

1 2

(2:12:14)

0

0

z ν ð1 ν − 1=2 2 ð1 − t2 Þ cosðz tÞdt Jν ðzÞ = pffiffiffi π Γ ν + 21 −1

1 Jν ðzÞ = π

ðπ 0

sinðπνÞ cosðz sin t − ν tÞdt − π

∞ ð

0

;

Re ν > −

1 2

e − ðz sinh t + νtÞ dt

(2:12:15)

;

Re z > 0

(2:12:16)

2.12 Integral Representation of the Bessel and Related Functions

1 Jν ðzÞ = π

π=2 ð

0

∞ ð

1 cosðz sin t − ν tÞdt + π

z>0

 π ν e − νt sin z cosh t − dt 2

35

(2:12:17)

0

; Re ν ≥ 0

 ν 2 z J−ν ðzÞ = pffiffiffi 2 1 πΓ ν + 2

"1

ð ð1 − t2 Þ cosðzt + πνÞdt 0

∞ ð

+ sinðπνÞ

#

(2:12:18)

e − z t ð1 + t2 Þν − 1=2 dt

;

Re z > 0 ;

Re ν > −

1 2

0

Y0 ðzÞ = −

2 Y0 ðzÞ = π 4 Y0 ðzÞ = 2 π Yν ðzÞ =

1 π 1 π

2 π

∞ ð

;

cosðz cosh tÞ dt

z>0 ;

jRe νj < 1

(2:12:19)

0

π=2 ð

cosðz cos tÞ ln½4zðsin tÞ2  dt

; z>0

(2:12:20)

0

ð1

sin − 1 t 4 sinðz tÞ pffiffiffiffiffiffiffiffiffiffi dt − 2 π 1 − t2

0

∞ ð

1

pffiffiffiffiffiffiffiffiffiffi lnðt + t2 − 1Þ pffiffiffiffiffiffiffiffiffiffi sinðz tÞ dt t2 − 1

z>0

(2:12:21)

ðπ sinðz sin t − ν tÞ dt − 0 ∞ ð

(2:12:22) e

− z sinh t

0

 ν 2 z2 Yν ðzÞ = pffiffiffi  π Γ ν + 21 Re ν > −

1 2

½e + cosðπνÞ e

;

−νt

 dt

; Re z > 0

" π=2

∞ ð

ð

sinðz sin tÞðcos tÞ2ν dt −

# e − z sinh t ðcosh tÞ2ν dt

0

0

;

 ν 2 z2 Yν ðzÞ = pffiffiffi  πΓ ν + 21 Re z >0

νt

Re z > 0

"1

∞ ð

ð ð1 − t Þ sinðztÞdt − 2

0

Re ν > −

0

1 2

#

(2:12:23)

e − zt ð1 + t2 Þν − 1=2 dt (2:12:24)

36

2 Properties of the Bessel and Related Functions

2 I0 ðzÞ = π

π=2 ð

coshðz cos tÞ dt

(2:12:25)

0

ðπ

I0 ðzÞ =

1 coshðz cos tÞ dt π

(2:12:26)

0

ð1 2 coshðz tÞ pffiffiffiffiffiffiffiffiffiffi dt I0 ðzÞ = π 1 − t2

(2:12:27)

0

1 I0 ðzÞ = π

ð1 −1

e−zt pffiffiffiffiffiffiffiffiffiffi dt 1 − t2

(2:12:28)

ð1 2 z pffiffiffiffiffiffiffiffiffiffi2 I0 ðzÞ = 1 − t coshðz tÞ dt π

(2:12:29)

ðπ 1 z cos t e cosðntÞ dt In ðzÞ = π

(2:12:30)

0

; n = 0, 1, 2, 3, . . .

0

π=2  ν ð 2 z2 Iν ðzÞ = pffiffiffi  coshðz cos tÞðsin tÞ2 ν dt π Γ ν + 21

(2:12:31)

0

Re ν > −

1 2

 z ν ðπ coshðz cos tÞðsin tÞ2 ν dt Iν ðzÞ = pffiffiffi 2 π Γ ν + 21

;

Re ν > −

1 2

(2:12:32)

0

Iν ðzÞ =

ðπ

∞ ð

0

0

1 z cos t sinðπνÞ e cosðνtÞ dt − π π

e − ðz cosh t + ν tÞ dt (2:12:33)

Re ν > 0 z ν ð1 ν − 1=2 2 e − z t ð1 − t2 Þ dt Iν ðzÞ = pffiffiffi π Γ ν + 21

;

−1

Re ν > −

(2:12:34)

1 2

z ν ð1 ν − 1=2 2 e − z t ð1 − t2 Þ Iν ðzÞ = pffiffiffi coshðz tÞdt π Γ ν + 21 −1

1 Re ν > − 2

(2:12:35)

37

2.12 Integral Representation of the Bessel and Related Functions

∞  ν ð 2 z sinðπ νÞ ν − 1=2 − z t I−ν ðzÞ = Iν ðzÞ + p2ffiffiffi  ðt2 − 1Þ e dt π Γ ν + 21

(2:12:36)

1

1 Re ν > − 2 2 sinðπνÞ I−ν ðzÞ = Iν ðzÞ + π

∞ ð

e − z cosh t coshðν tÞ dt

;

Re ν > 0

(2:12:37)

0

" ð1 # ∞ ð 2 2 1 ν − 1=2 zt 2 ν − 1=2 − zt 2 I− ν ðzÞ = pffiffiffi e ð1 − t Þ dt + sinðπνÞ e ðt − 1Þ dt πΓðν + 21Þ 2  z ν

−1

Re z > 0

1

1 ; Re ν > − 2 (2:12:38)

∞ ð

K0 ðzÞ = 0 ∞ ð

K0 ðzÞ = 0 ∞ ð

K0 ðzÞ =

pffiffiffiffiffiffiffiffi 2 e−z 1+t pffiffiffiffiffiffiffiffiffiffi dt 1 + t2

;

K0 ðzÞ = 0 ∞ ð

K0 ðzÞ = 0

(2:12:39)

cosðztÞ pffiffiffiffiffiffiffiffiffiffi dt 1 + t2 cosðz sinh tÞ dt

0 ∞ ð

z>0

(2:12:40)

; z>0

(2:12:41)

t J0 ðztÞ dt 1 + t2

(2:12:42)

Y0 ðztÞ dt 1 + t2

(2:12:43)

∞ ð

K0 ðzÞ = z

t e − z cosh t sinh t dt

(2:12:44)

0 ∞ ð

Kν ðzÞ =

e − z cosh t coshð νtÞ dt

;

Re z > 0

(2:12:45)

0

1 Kν ðzÞ = 2

∞ ð

−∞

e − z cosh t − ν t dt

;

Re z > 0

(2:12:46)

38

2 Properties of the Bessel and Related Functions

pffiffiffi z ν ∞ ð π Kν ðzÞ =  2 1 e − z cosh t ðsinh tÞ2ν dt Γ ν+ 2

(2:12:47)

0

1 Re z > 0 ; Re ν > − 2   ν ∞ ð Γ ν + 21 z2 cosðz sinh tÞ pffiffiffi dt Kν ðzÞ = π ðcosh tÞ2 ν

(2:12:48)

0

1 Re z > 0 ; Re ν > − 2  ð ν∞ 1 Γ ν + 2 ð2 z Þ cos t pffiffiffi dt Kν ðzÞ = π ð1 + t2 Þν + 1=2

(2:12:49)

0

Re z > 0 Kν ðzÞ =

1 2

∞ ð

1 Re ν > − 2

;

tν − 1 e − z ð t + 1=tÞ=2 dt

;

Re z > 0 ;

Re ν > −

1 2

(2:12:50)

0 ∞ ð 1 z ν 2 t − ðν + 1Þ e − ðt + z =4 tÞ dt Kν ðzÞ = 2 2 0

(2:12:51)

1 Re ν > − 2

Re z > 0 ;

pffiffiffi z ν ∞ ð π ν − 1=2 − z t e dt Kν ðzÞ =  2 1 ðt2 − 1Þ Γ ν+ 2

(2:12:52)

1 Re z > 0 ; Re ν > − 2 ∞ pffiffiffiffi   π −z ð t 2z e ν − 1=2  e − t dt t 1+ Kν ðzÞ = 2z Γ ν + 21

(2:12:53)

1 Re z > 0 ; Re ν > − 2 pffiffiπ ν − z ∞  ð ν − 1=2 ze t2 t + e − z t dt Kν ðzÞ = 2 2 Γ ν + 21

(2:12:54)

1

0

0

Re z > 0

1 Re ν > − 2

;

pffiffiffi z ν − z ð1 π 2 e  tz − 1 ½ðln tÞ2 − 2 ln t ν − 1=2 dt Kν ðzÞ = Γ ν + 21 0

(2:12:55)

2.12 Integral Representation of the Bessel and Related Functions

2 H 0 ðzÞ = π H 0 ðzÞ =

ð1 0

sinðz tÞ pffiffiffiffiffiffiffiffiffiffi dt 1 − t2

1 2 Y0 ðzÞ + π π

∞ ð

39

(2:12:56)

e − z sinh t dt

(2:12:57)

0

π=2  ν ð 2 z2 H ν ðzÞ = pffiffiffi  sinðz cos tÞðsin tÞ2 ν dt π Γ ν + 21

;

Re ν > −

1 2

(2:12:58)

0

 ν ð1 2 z ν − 1=2 sinðz tÞdt H ν ðzÞ = pffiffiffi 2 1 ð1 − t2 Þ πΓ ν+ 2

;

Re ν > −

1 2

(2:12:59)

0

∞  ν ð 2 z ν − 1=2 dt H ν ðzÞ = Yν ðzÞ + pffiffiffi 2 1 e − z t ð1 + t2 Þ πΓ ν+ 2

; Re ν > 0

(2:12:60)

0

L0 ðzÞ = I0 ðzÞ−

L0 ðzÞ = I0 ðzÞ−

2 π 2 π

∞ ð

0 ∞ ð

0

sinðz sinh tÞ dt

(2:12:61)

sinðz tÞ pffiffiffiffiffiffiffiffiffiffi dt 1 + t2

(2:12:62)

π=2  ν ð 2 z2 Lν ðzÞ = pffiffiffi  sinhðz cos tÞðsin tÞ2 ν dt π Γ ν + 21 0

Re ν > −

(2:12:63)

1 2

 ν ð1 2 z2 ν − 1=2 e − z t ð1 − t2 Þ Lν ðzÞ = Iν ðzÞ− pffiffiffi  dt 1 πΓ ν+ 2 0

(2:12:64)

1 Re ν > − 2 sinðπ νÞ J ν ðzÞ = Jν ðzÞ + π Eν ðzÞ = − Yν ðzÞ− Re z > 0

1 π

∞ ð

0



∞ ð

e − z sinh t − ν t d t

;

Re z > 0

(2:12:65)

0

eν t + e − ν t cosðπ νÞ e − z sinh t d t

(2:12:66)

40

2 Properties of the Bessel and Related Functions

1 ν sinðπ νÞ

s−1, ν ðzÞ = −

cosðz sin tÞ cosðν tÞ dt

1  ν sin πν 2

s−1, ν ðzÞ = −

1 sinðπ νÞ

ðπ cosðz sin tÞ sinðν tÞ dt

(2:12:68)

0

π=2 ð

cosðz cos tÞ cosðνtÞ dt

(2:12:69)

0

ðπ sinðz sin tÞ cosðν tÞ dt

(2:12:70)

0

1  s0, ν ðzÞ = ν cos πν 2

π=2 ð

cosðz cos tÞ cosðνtÞ dt

π=2 ð

(2:12:71)

0

1 s0, ν ðzÞ = − ½1 + cosðπ νÞ pffiffiffi sν, ν ðzÞ = π zν

(2:12:67)

0

1 ν ½1 − cosðπ νÞ

s−1, ν ðzÞ = −

s0, ν ðzÞ =

ðπ

ðπ sinðz sin tÞ cosðνtÞ dt

(2:12:72)

0

ðcos tÞ2 ν sinðz sin tÞ dt

; Reν > −

1 2

(2:12:73)

0

1 sμ − 1=2, 1=2 ðzÞ = pffiffiffi z

ðz

tμ − 1 sinðz − tÞ dt

;

μ>0

(2:12:74)

0

sμ, ν ðzÞ = 2μ Γð π=2 ð

IðzÞ =

μ − ν + 1 z ðμ + ν + 1Þ=2 IðzÞ Þ 2 2

ðcos tÞλ + ν ðsin tÞðν − μ +

1Þ=2

Jðμ − ν + 1Þ=2 ðz sin tÞ dt

(2:12:75)

0

Reðμ + νÞ > − 1 S−2, 1 ðzÞ =

1 2

∞ ð

t e − z sinh t sinh t dt =

1 2

0 ∞ ð

∞ ð

0

t e − z t sinh − 1 t pffiffiffiffiffiffiffiffiffiffi dt 1 + t2

(2:12:76)

t e − z sinh t dt

S−1, 0 ðzÞ = 0 ∞ ð

= 0

e − z t sinh − 1 t z pffiffiffiffiffiffiffiffiffiffi dt = 2 1 + t2

(2:12:77)

∞ ð

e 0

−zt

−1 2

½sinh t dt

2.12 Integral Representation of the Bessel and Related Functions

S0, 1=3 ðzÞ =

pffiffiffi ∞ ð 3 z K1=3 ðtÞ dt z2 + t2 π

41

(2:12:78)

0

∞ ð

S0, ν ðzÞ =

e − z sinh t coshðνtÞ dt

;

Re z > 0

(2:12:79)

0

z S0, ν ðzÞ = ν

∞ ð

; Re z > 0

(2:12:80)

e − z sinh t coshðνtÞ cosh t dt

;

(2:12:81)

0 ∞ ð

S1, ν ðzÞ = z

Sν, ν ðzÞ = zν

e − z sinh t sinhðνtÞ cosh t dt

0 ∞ ð

e − z sinh t ðcosh tÞ2ν dt

;

Re z > 0

Re z > 0

(2:12:82)

0

ð2zÞμ + 1 Sμ, ν ðzÞ = 1 − μ − ν 1 − μ + ν Γ 2 Γ 2

∞ ð

0

x − μ Kν ðxÞ dt z2 + t2

(2:12:83)

Reðμ ± νÞ < 1 Sμ, ν ðzÞ = z

μ+1

∞ ð

t e − z t 2 F1

  1−μ+ν 1−μ−ν 3 2 , ; ; −t dt 2 2 2

0

(2:12:84)

Re z > 0 ð1 pffiffiffi 2 cosðz tÞ coshðz tÞ pffiffiffiffiffiffiffiffiffiffi berð 2 zÞ = dt π 1 − t2

(2:12:85)

0

ðπ pffiffiffi ð− 1Þn cosðz sin t − n tÞ coshðz sin tÞ dt bern ð 2 zÞ = π 0

(2:12:86)

n = 0, 1, 2, 3, ... ðπ pffiffiffi 1 ½cosðπ νÞ cosðz sin t − νtÞ coshðz sin tÞ berν ð 2 zÞ = π 0

− sinðπ νÞ sinðz sin t − νtÞ sinhðz sin tÞ dt ∞ ð sinðπ νÞ e − ν t − z sinh t cosðz sinh t + π νÞ dt − π 0

(2:12:87)

42

2 Properties of the Bessel and Related Functions

ð1 pffiffiffi 2 sinðz tÞ sinhðz tÞ pffiffiffiffiffiffiffiffiffiffi beið 2 zÞ = dt π 1 − t2

(2:12:88)

0

ðπ pffiffiffi ð− 1Þn sinðz sin t − ntÞ sinhðz sin tÞ dt bein ð 2 zÞ = π 0

(2:12:89)

n = 0, 1, 2, 3, ... ðπ pffiffiffi 1 ½cosðπ νÞ sinðz sin t − νtÞ sinhðz sin tÞ beiν ð 2 zÞ = π 0

+ sinðπ νÞ cosðz sin t − νtÞ coshðz sin tÞ dt ∞ ð sinðπ νÞ − e − ν t − z sinh t sinðz sinh t + π νÞ dt π

(2:12:90)

0

∞ ð

pffiffiffi kerð 2 zÞ =

e − z cosh t cosðz cosh tÞdt

(2:12:91)

0 ∞ ð

z2 8

kerðzÞ = −

e−z

2 t=4

Ci

  1 dt t

(2:12:92)

0 ∞ ð

t J0 ðz tÞ dt 1 + t4 3

kerðzÞ = 0

kerðzÞ =

z 4

(2:12:93)

∞ ð

lnð1 + t4 Þ J1 ðz tÞ dt

(2:12:94)

0 ∞ ð  pffiffiffi π ν kerν ð 2 zÞ = e − z cosh t cos z cosh t + coshðνtÞdt 2 0 ∞ ð

pffiffiffi keið 2 zÞ = −

(2:12:95)

e − z cosh t sinðz cosh tÞdt

(2:12:96)

    1 π − dt Si t 2

(2:12:97)

0

keiðzÞ =

z2 8

∞ ð

0 ∞ ð

keiðzÞ = − 0

z keiðzÞ = − 2

e−z

2 t=4

t J0 ðz tÞ dt 1 + t4

∞ ð

0

tan − 1 ðt2 Þ J1 ðz tÞ dt

(2:12:98)

(2:12:99)

43

2.13 Differential Equations Reducible to the Bessel Differential Equations

∞ ð  pffiffiffi π ν keiν ð 2 zÞ = − e − z cosh t sin z cosh t + coshðνtÞdt 2

(2:12:100)

0

Ji0 ðzÞ =

2 π

∞ ð

siðz cosh tÞdt

(2:12:101)

0

z ðz J ðtÞ− 1 0 Ji0 ðzÞ = γ + ln + dt 2 t

(2:12:102)

0

ðz Ji0 ðzÞ = γ + J0 ðzÞ ln z − ln 2 +

J0 ðtÞ ln t dt

(2:12:103)

0

Ji2n ðzÞ =

ðπ 1 cosð2ntÞ ciðz sin tÞdt π

(2:12:104)

0

ðπ 1 sin½ð2n + 1Þt siðz sin tÞ dt Ji2n + 1 ðzÞ = π Yi0 ðzÞ = −

Kiν ðzÞ =

2 π

1 ΓðνÞ

0 ∞ ð

ciðz cosh tÞdt 0 ∞ ð

Kiν ðzÞ =

(2:12:106)

ðt − zÞν − 1 K0 ðtÞ dt

(2:12:107)

z

Re z > 0 ∞ ð

(2:12:105)

;

Re ν > 0

e − z cosh t dt ðcosh tÞν

; Re z > 0

; Re ν > 0

(2:12:108)

0

2.13 Differential Equations Reducible to the Bessel Differential Equations Differential equations presented here were collected from many sources in the literature, mainly from [7–9, 13, 14, 16, 18, 20, 21, 23, 24, 30–32]. For the readers convenience, this list of differential equations includes not only these which are of less or more general character, but also particular cases having only different coefficients. Most of the solutions expressible in terms of the Bessel functions are for differential equations that contain power functions, much less exist for exponential or other elementary functions.

44

2 Properties of the Bessel and Related Functions

The following notation is used to express solutions of differential equations Cν ðzÞ = A1 Jν ðzÞ + A2 Yν ðzÞ and Zν ðzÞ = A1 Iν ðzÞ + A2 Kν ðzÞ   pffiffiffi d2 wðzÞ 2 3=2 (2:13:1) + z wðzÞ = 0 ; wðzÞ = z C z 1=3 d z2 3   pffiffiffi d2 wðzÞ 2 3=2 − z wðzÞ = 0 ; wðzÞ = z C i z 1=3 d z2 3 (2:13:2) wðzÞ = A1 AiðzÞ + A2 BiðzÞ α pffiffiffi wðzÞ = z C1=4 z2 2  2 3 pffiffiffi d2 wðzÞ αz 4 4 + α z wðzÞ = 0 ; wðzÞ = z C1=6 3 d z2   pffiffiffi d2 wðzÞ 1 4 3=4 + pffiffiffi wðzÞ = 0 ; wðzÞ = z C2=3 z d z2 3 z   pffiffiffi d2 wðzÞ 1 4 3=4 p ffiffi ffi wðzÞ = 0 ; wðzÞ = z C − i z 2=3 d z2 3 z d2 wðzÞ + α2 z2 wðzÞ = 0 d z2

;

(2:13:3) (2:13:4)

(2:13:5)

(2:13:6)

d2 wðzÞ 1 + wðzÞ = 0 ; d z2 z

pffiffiffi pffiffiffi wðzÞ = z C1 ð2 zÞ

(2:13:7)

d2 wðzÞ 1 − wðzÞ = 0 d z2 z

pffiffiffi pffiffiffi wðzÞ = z Z1 ð2 zÞ

(2:13:8)

;

d2 wðzÞ + α2 z − 3=2 wðzÞ = 0 ; d z2

pffiffiffi wðzÞ = z C2 ð4αz1=4 Þ

  pffiffiffi 2 αzβ=2 + 1 wðzÞ = z C1=ðβ + 2Þ β+2   pffiffiffi d2 wðzÞ 2 αz1 − β=2 2 2−β + α z wðzÞ = 0 ; wðzÞ = z C1=ð2 − βÞ 2−β d z2  β − 1=2  pffiffiffi d2 wðzÞ iαz 2 2β−2 − α z wðzÞ = 0 ; wðzÞ = z C 1=ð2 β − 2Þ 2 β−1 dz   pffiffiffi pffiffiffi d2 wðzÞ 1 z2 wðzÞ = 0 ; wðzÞ = z C0 ð2 α zÞ + α2 z + 2 dz 4   d2 wðzÞ 3 2 2 z2 wðzÞ = 0 ; wðzÞ = z C1 ðα zÞ + α z − d z2 4 d2 wðzÞ + α2 zβ + 2 wðzÞ = 0 d z2

z2

;

d2 wðzÞ + ðα2 z2 − 2ÞwðzÞ = 0 d z2

;

wðzÞ =

1 C3=2 ðα zÞ z

(2:13:9) (2:13:10)

(2:13:11)

(2:13:12)

(2:13:13)

(2:13:14)

(2:13:15)

2.13 Differential Equations Reducible to the Bessel Differential Equations

z2

  d2 wðzÞ 15 2 2 wðzÞ = 0 ; + α z − d z2 4

z2

d2 wðzÞ + ðα2 z + βÞwðzÞ = 0 d z2

z2

  d2 wðzÞ 1 2 2 2 wðzÞ = 0 ; + α z − α + d z2 4

;

pffiffiffi wðzÞ = z C2 ðαzÞ

wðzÞ =

(2:13:17)

pffiffiffi wðzÞ = z Cβ ðα zÞ

(2:13:18)

d2 wðzÞ − ½α2 z2 + βð1 − βÞ wðzÞ = 0 d z2 pffiffiffi wðzÞ = z Cβ − 1=2 ði α zÞ     2 pffiffiffi 1 2 α 3=2 2 d wðzÞ 2 3 z wðzÞ = 0 ; wðzÞ = z C0 + αz + z d z2 4 3 d2 wðzÞ z + ðα2 z3 − βÞwðzÞ = 0 d z2

;

z2

d2 wðzÞ + ðα2 z4 + 4ÞwðzÞ = 0 ; d z2

  d2 wðzÞ β 2 4 wðzÞ = 0 z + αz − d z2 4 2

(2:13:19)

(2:13:20)

  pffiffiffi 2 α 3=2 p ffiffiffiffiffiffiffiffiffi wðzÞ = z C1 1 + 4 β z 3 3

(2:13:21)

pffiffiffi α wðzÞ = z C0 z2 2

(2:13:22)

 pffiffiffi ffi α z2 ; wðzÞ = z C1pffiffiffiffiffiffi 2 1+β 2

  pffiffiffi d2 wðzÞ 2 α 5=2 2 5 z + ðα z − 2ÞwðzÞ = 0 ; wðzÞ = z C3=5 z d z2 5    2  pffiffiffi d2 wðzÞ α 1 2α pffiffiffi z2 wðzÞ = 0 ; wðzÞ = z C + + 0 2 z 4 dz z     pffiffiffi d2 wðzÞ 1 2 α 3=2 2 3 z2 wðzÞ = 0 ; wðzÞ = z C + α z + z 0 d z2 4 3   d2 wðzÞ n2 + 2 n z2 wðzÞ = 0 + α2 z n + 2 − 2 dz 4   pffiffiffi 2 α n=2 + 1 wðzÞ = z Cðn + 1Þ=ðn + 2Þ z n+2   d2 wðzÞ n2 + 4n + 3 2 n+2 z2 wðzÞ = 0 + α z − d z2 4   pffiffiffi 2 α n=2 + 1 wðzÞ = z C1 z n+2 2

(2:13:16)

pffiffiffi pffiffiffi z Cpffiffiffiffiffiffiffiffi ð2 α zÞ 1 − 4β

z2

2

45

(2:13:23)

(2:13:24)

(2:13:25)

(2:13:26)

(2:13:27)

(2:13:28)

46

2 Properties of the Bessel and Related Functions

  d2 wðzÞ 1 2 β+2 wðzÞ = 0 + α z + dz2 4   pffiffiffi 2 α β=2 + 1 wðzÞ = z C0 z β+2

z2

d2 wðzÞ + ðα2 zβ + 2 + λ2 ÞwðzÞ = 0 ; α≠0 ; dz2   pffiffiffi 2α ðβ + 2Þ=2 ffiffiffiffiffiffiffiffiffiffi p wðzÞ = z C z 1 − 4 λ2 =ðβ + 2Þ β + 2   2 1 2 d wðzÞ 2 2 α 2 2 z wðzÞ = 0 + α β z −α λ + dz2 4 pffiffiffi wðzÞ = z Cλ ðβzα Þ ! 2 1 − α2 β2 2 d wðzÞ 2 2 α z + αλz + wðzÞ = 0 4 dz2 pffiffiffi wðzÞ = z Cλ ð2λzα=2 Þ   d2 wðzÞ dwðzÞ 1 2 n wðzÞ = 0 + z + + α d z2 dz 4   pffiffiffi − z=2 2α n=2 + 1 wðzÞ = z e C1=ðn + 2Þ z n+2   d2 wðzÞ dwðzÞ 1 2 n wðzÞ = 0 − z + + α d z2 dz 4   pffiffiffi z=2 2α n=2 + 1 wðzÞ = z e C1=ðn + 2Þ z n+2 z2

d2 wðzÞ dwðzÞ +2 + ðα2 zn + 1ÞwðzÞ = 0 dz2 dz   pffiffiffi − z 2 α n=2 + 1 wðzÞ = z e C1=ðn + 2Þ z n+2 d2 wðzÞ dwðzÞ −2 + ðα2 zn + 1ÞwðzÞ = 0 2 dz dz   pffiffiffi 2α n=2 + 1 wðzÞ = z ez C1=ðn + 2Þ z n+2   d2 wðzÞ dwðzÞ α2 2 wðzÞ = 0 + α + β + 4 d z2 dz pffiffiffi wðzÞ = z e − α z=2 C1=2 ðβzÞ

(2:13:29)

β≠ − 2 (2:13:30)

(2:13:31)

(2:13:32)

(2:13:33)

(2:13:34)

(2:13:35)

(2:13:36)

(2:13:37)

2.13 Differential Equations Reducible to the Bessel Differential Equations

 2  d2 wðzÞ dwðzÞ α 2 + α z wðzÞ = 0 + β + 4 d z2 dz   pffiffiffi 2 β 3=2 wðzÞ = z e − α z=2 C1=3 z 3   d2 wðzÞ dwðzÞ α2 2 n wðzÞ = 0 +α + β z + 4 d z2 dz   2β n=2 + 1 wðzÞ = z e − α z=2 C0 z n+2 z

d2 wðzÞ 1 dwðzÞ 1 + + wðzÞ = 0 dz2 2 dz 4

z

d2 wðzÞ dwðzÞ − + z wðzÞ = 0 dz2 dz

pffiffiffi wðzÞ = z1=4 C1=2 ð zÞ

; ;

pffiffiffi wðzÞ = z C1 ð zÞ

d2 wðzÞ dwðzÞ α2 + β wðzÞ = 0 + 4 d z2 dz pffiffiffi wðzÞ = zð1 − βÞ=2 C1 − β ðα zÞ z

z

d2 wðzÞ dwðzÞ −2β − α2 z wðzÞ = 0 d z2 dz

wðzÞ = z

β + 1=2

Zβ +

47

(2:13:38)

(2:13:39)

(2:13:40) (2:13:41)

(2:13:42)

(2:13:43)

1=2 ðα zÞ

d2 wðzÞ dwðzÞ + ð1 + βÞ + wðzÞ = 0 2 dz dz pffiffiffi wðzÞ = z − β=2 Cβ ð2 zÞ

(2:13:44)

d2 wðzÞ dwðzÞ + ð1 + βÞ − wðzÞ = 0 d z2 dz pffiffiffi wðzÞ = z − β=2 Zβ ð2 zÞ

(2:13:45)

d2 wðzÞ dwðzÞ 1 + ð1 − βÞ + wðzÞ = 0 2 dz dz 4 pffiffiffi β=2 wðzÞ = z Cβ ð zÞ

(2:13:46)

z

z

z

z

d2 wðzÞ dwðzÞ + ð1 + 2βÞ − z wðzÞ = 0 d z2 dz

(2:13:47)

wðzÞ = z − β Zβ ðzÞ z

d2 wðzÞ dwðzÞ + ð1 − 2βÞ + z wðzÞ = 0 d z2 dz β

wðzÞ = z Cβ ðzÞ

(2:13:48)

48

2 Properties of the Bessel and Related Functions

z2

d2 wðzÞ dwðzÞ + 2z + α2 z wðzÞ = 0 d z2 dz

pffiffiffi ; wðzÞ = C1 ð2 α zÞ

d2 wðzÞ dwðzÞ + ðβ + 1Þz + α2 z wðzÞ = 0 2 dz dz pffiffiffi wðzÞ = z − β=2 Cβ ð2 α zÞ z2

(2:13:49)

(2:13:50)

! d2 wðzÞ z dwðzÞ β2 1 2 z + + + α z− wðzÞ = 0 4 16 dz2 2 dz pffiffiffi wðzÞ = z1=4 Cβ ð2 α zÞ   pffiffiffi d2 wðzÞ dwðzÞ 1 2 wðzÞ = 0 ; wðzÞ = C1 ð2 α zÞ + z z − z2 + α d z2 dz 4 2

z2

d2 wðzÞ dwðzÞ 1 +z + ðz − β2 ÞwðzÞ = 0 d z2 dz 4

pffiffiffi ; wðzÞ = Cβ ð zÞ

(2:13:51)

(2:13:52)

(2:13:53)

! d2 wðzÞ dwðzÞ β2 2 +z z wðzÞ = 0 + α z− 4 d z2 dz pffiffiffi wðzÞ = Cβ ð2 α zÞ

(2:13:54)

d2 wðzÞ dwðzÞ −z + ðα2 z + β2 ÞwðzÞ = 0 dz2 dz pffiffiffi wðzÞ = z C pffiffiffiffiffiffiffiffi ð2 α zÞ

(2:13:55)

2

z2

2

1 − β2

d2 wðzÞ dwðzÞ −z + ðα2 z + 1ÞwðzÞ = 0 2 dz dz pffiffiffi wðzÞ = z C0 ð2 α zÞ ! 2 dwðzÞ β2 2 d wðzÞ 2 z −z wðzÞ = 0 + α z+ 1− 4 dz2 dz pffiffiffi wðzÞ = z Cβ ð2 α zÞ   2 dwðzÞ 1 2 d wðzÞ 2 z wðzÞ = 0 +2z + α z+ d z2 dz 4 z2

pffiffiffi 1 wðzÞ = pffiffiffi C0 ð2α zÞ z

(2:13:56)

(2:13:57)

(2:13:58)

2.13 Differential Equations Reducible to the Bessel Differential Equations

d2 wðzÞ dwðzÞ +2z + ðα2 z + β2 ÞwðzÞ = 0 dz2 dz pffiffiffi 1 wðzÞ = pffiffiffi Cpffiffiffiffiffiffiffiffiffiffi2ffi ð2 α zÞ 1−4β z ! 2 dwðzÞ 1 − β2 2 d wðzÞ 2 +2z wðzÞ = 0 z + α z+ 4 dz2 dz

49

z2

pffiffiffi 1 wðzÞ = pffiffiffi Cβ ð2 α zÞ z   d2 wðzÞ dwðzÞ 9 2 z wðzÞ = 0 −2z + α z+ dz2 dz 4 pffiffiffi wðzÞ = z3=2 C0 ð2α zÞ

(2:13:59)

(2:13:60)

2

d2 wðzÞ dwðzÞ −2z + ðα2 z + 2ÞwðzÞ = 0 dz2 dz pffiffiffi wðzÞ = z3=2 C1 ð2 α zÞ ! 2 dwðzÞ 9 − β2 2 d wðzÞ 2 z −2z wðzÞ = 0 + α z+ 4 dz2 dz pffiffiffi wðzÞ = z3=2 Cβ ð2 α zÞ " # 2 dwðzÞ ðβ + 1Þ2 2 d wðzÞ 2 − βz wðzÞ = 0 + αz+ z 4 dz2 dz z2

wðzÞ = z

ðβ + 1Þ=2

(2:13:61)

(2:13:62)

(2:13:63)

(2:13:64)

pffiffiffi C0 ð2α zÞ

d2 wðzÞ dwðzÞ − βz + ðα2 z + βÞwðzÞ = 0 dz2 dz pffiffiffi wðzÞ = zðβ + 1Þ=2 Cβ − 1 ð2 α zÞ ! 2 dwðzÞ β2 2 d wðzÞ 2 z + ðβ + 1Þ z wðzÞ = 0 + α z+ 4 dz2 dz pffiffiffi wðzÞ = z − β=2 C0 ð2 α zÞ ! 2 dwðzÞ 3β2 2 d wðzÞ + ðβ + 1Þ z wðzÞ = 0 + z− z 4 dz2 dz z2

pffiffiffi wðzÞ = z − β=2 C2 β ð2 zÞ

(2:13:65)

(2:13:66)

(2:13:67)

50

2 Properties of the Bessel and Related Functions

 2 dwðzÞ z2 d dzwðzÞ + ðβ + 1Þz + α2 z + 2 dz pffiffiffi wðzÞ = z − β=2 C1 ð2α zÞ

β2 − 1 4



wðzÞ = 0

(2:13:68)

d2 wðzÞ dwðzÞ − ðβ + 1Þ z + ðα2 z + λ2 ÞwðzÞ = 0 dz2 dz pffiffiffi wðzÞ = zβ=2 + 1 Cpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 α zÞ 2 z2

(2:13:69)

ðβ + 2Þ − 4λ2

d2 wðzÞ dwðzÞ − ðβ + 1Þ z + ðα2 z + β + 1ÞwðzÞ = 0 dz2 dz pffiffiffi wðzÞ = zβ=2 + 1 Cν ð2 α zÞ " # 2 dwðzÞ ðβ + 2Þ2 2 d wðzÞ 2 z − ðβ + 1Þ z wðzÞ = 0 + αz+ 4 dz2 dz pffiffiffi wðzÞ = zβ=2 + 1 C0 ð2 α zÞ z2

z2

d2 wðzÞ dwðzÞ + ðβ + 1Þ z + α2 z2 wðzÞ = 0 dz2 dz

wðzÞ = z

−β

d2 wðzÞ dwðzÞ + ð2β + 1Þz + α2 z2 wðzÞ = 0 ; d z2 dz   d2 wðzÞ z dwðzÞ 1 2 2 wðzÞ = 0 + z + z2 + α dz2 2 dz 16 wðzÞ = z z2

wðzÞ = z z2

(2:13:72)

wðzÞ = Cβ ðα zÞ

(2:13:73)

(2:13:74)

C0 ðα zÞ

  d2 wðzÞ z dwðzÞ 1 2 2 2 wðzÞ = 0 + z − β + + α dz2 2 dz 16 1=4

(2:13:71)

Cβ=2 ðα zÞ

z2

1=4

(2:13:70)

(2:13:75)

Cβ ðα zÞ

d2 wðzÞ dwðzÞ +z + 4 ðz2 − 1ÞwðzÞ = 0 dz2 dz

(2:13:76)

wðzÞ = C2 ð2zÞ   d2 wðzÞ dwðzÞ 1 2 z wðzÞ = 0 +z + z − dz2 dz 16 2

wðzÞ = C1=4 ðzÞ

(2:13:77)

2.13 Differential Equations Reducible to the Bessel Differential Equations

 d2 wðzÞ dwðzÞ + z − 3z2 + dz2 dz pffiffiffi wðzÞ = Z1=3 ð 3 zÞ  d2 wðzÞ dwðzÞ z2 + z + 3z2 − dz2 dz z2

 1 wðzÞ = 0 4

51

(2:13:78)

 1 wðzÞ = 0 ; 4

pffiffiffi wðzÞ = C1=3 ð 3 zÞ

(2:13:79)

d2 wðzÞ dwðzÞ +z (2:13:80) + ðα2 z2 − 1ÞwðzÞ = 0 ; wðzÞ = C1 ðα zÞ d z2 dz " # 2 dwðzÞ ð2 n + 1Þ2 2 d wðzÞ 2 2 z +z wðzÞ = 0 ; wðzÞ = Cn + 1=2 ðα zÞ + αz − 4 dz2 dz z2

(2:13:81) z2

d2 wðzÞ dwðzÞ +z + ðα2 z2 − ν2 ÞwðzÞ = 0 d z2 dz

z2

d2 wðzÞ dwðzÞ −z + ðα2 z2 + 1ÞwðzÞ = 0 d z2 dz

; ;

d2 wðzÞ dwðzÞ −z + ðα2 z2 + β2 ÞwðzÞ = 0 2 dz dz wðzÞ = Cpffiffiffiffiffiffiffiffi ðα zÞ

wðzÞ = Cν ðα zÞ

(2:13:82)

wðzÞ = z C0 ðαzÞ

(2:13:83)

z2

(2:13:84)

1 − β2

z2

d2 wðzÞ dwðzÞ −z + ðα2 z2 − β2 + 1Þ wðzÞ = 0 dz2 dz

(2:13:85)

wðzÞ = z Cβ ðαzÞ   d2 wðzÞ dwðzÞ 1 2 2 z wðzÞ = 0 + 2z + αz − dz2 dz 4 2

1 wðzÞ = pffiffiffi C0 ðα zÞ z

(2:13:86)

d2 wðzÞ dwðzÞ + 2z + ðα2 z2 − 2Þ wðzÞ = 0 2 dz dz 1 wðzÞ = pffiffiffi C3=2 ðα zÞ z

(2:13:87)

d2 wðzÞ dwðzÞ + 2z + ðα2 z2 − β2 Þ wðzÞ = 0 2 dz dz 1 wðzÞ = pffiffiffi C1pffiffiffiffiffiffiffiffiffiffi2ffi ðαzÞ z 2 1−4β

(2:13:88)

z2

z2

52

2 Properties of the Bessel and Related Functions

d2 wðzÞ dwðzÞ + 2z + ðα2 z2 − β2 Þ wðzÞ = 0 dz2 dz 1 wðzÞ = pffiffiffi C1pffiffiffiffiffiffiffiffiffiffi2ffi ðα zÞ z 2 1−4β

(2:13:89)

d2 wðzÞ dwðzÞ + 2z + ðα2 z2 − 4 β2 + 1ÞwðzÞ = 0 dz2 dz 1 wðzÞ = pffiffiffi Cβ ðα zÞ z

(2:13:90)

d2 wðzÞ dwðzÞ −2z + β2 ðα2 z2 + β2 ÞwðzÞ = 0 dz2 dz wðzÞ = z3=2 C pffiffiffiffiffiffiffiffiffiffiffi ðα zÞ

(2:13:91)

z2

z2

z2

9 − 4 β2

1 2

z2

  d2 wðzÞ dwðzÞ 9 2 2 2 2 wðzÞ = 0 − 2 z α z − β + + β dz2 dz 4

wðzÞ = z

3=2

(2:13:92)

Cβ ðα zÞ

d2 wðzÞ dwðzÞ + 3z + ðα2 z2 + 1ÞwðzÞ = 0 dz2 dz 1 wðzÞ = C0 ðα zÞ z ! 2 dwðzÞ β2 2 d wðzÞ 2 2 + ðβ + 1Þ z wðzÞ = 0 z + αz + 4 dz2 dz z2

(2:13:93)

(2:13:94)

wðzÞ = z − β=2 C0 ðα zÞ ! d2 wðzÞ dwðzÞ 3β2 2 2 + ðβ + 1Þ z wðzÞ = 0 + αz − z 4 dz2 dz 2

(2:13:95)

wðzÞ = z − β=2 Cβ ðα zÞ d2 wðzÞ dwðzÞ + ðβ + 1Þ z + ðα2 z 2 + λ2 ÞwðzÞ = 0 dz2 dz wðzÞ = z − β=2 C pffiffiffiffiffiffiffiffiffiffiffiffi ðα zÞ z2

1 2

(2:13:96)

β2 − 4 λ2

" # d2 wðzÞ dwðzÞ ðβ + 2Þ2 2 2 z − ðβ + 1Þ z wðzÞ = 0 + αz + 4 dz2 dz 2

wðzÞ = z

β=2 + 1

C0 ðα zÞ

(2:13:97)

2.13 Differential Equations Reducible to the Bessel Differential Equations

d2 wðzÞ dwðzÞ − ðβ + 1Þ z + ðα2 z 2 + λ2 ÞwðzÞ = 0 dz2 dz ffi ðα zÞ wðzÞ = zβ=2 + 1 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

53

z2

1 2

z2

(2:13:98)

ðβ + 2Þ − 4 λ2

d2 wðzÞ dwðzÞ + ð1 + 2βÞ z + ðβ2 + z2 ÞwðzÞ = 0 dz2 dz

(2:13:99)

wðzÞ = z − β C0 ðzÞ z2

d2 wðzÞ dwðzÞ +z + α2 z3 wðzÞ = 0 d z2 dz

;

wðzÞ = C0

  2 α 3=2 z 3

d2 wðzÞ dwðzÞ −2z + α2 z3 wðzÞ = 0 dz2 dz   2α 3=2 wðzÞ = z3=2 C1 z 3   2 z dwðzÞ 1 2 d wðzÞ 2 3 wðzÞ = 0 + z + α z + dz2 2 dz 16   2 α 3=2 wðzÞ = z1=4 C0 z 3 ! 2 z dwðzÞ 1 − 36β2 2 d wðzÞ 2 3 z + wðzÞ = 0 + α z + 16 dz2 2 dz   2 α 3=2 wðzÞ = z1=4 Cβ z 3   d2 wðzÞ dwðzÞ 9 2 3 wðzÞ = 0 + z z − z2 + α dz2 dz 4   2 α 3=2 wðzÞ = C1 z 3

(2:13:100)

z2

(2:13:101)

(2:13:102)

(2:13:103)

(2:13:104)

d2 wðzÞ dwðzÞ +z + ðα2 z3 − β2 ÞwðzÞ = 0 2 dz dz   2 α 3=2 wðzÞ = C2 β=3 z 3

(2:13:105)

d2 wðzÞ dwðzÞ −z + ðα2 z3 − 1ÞwðzÞ = 0 dz2 dz   2 α 3=2 wðzÞ = z C0 z 3

(2:13:106)

z2

z2

54

2 Properties of the Bessel and Related Functions

  d2 wðzÞ dwðzÞ 5 2 3 wðzÞ = 0 − z z − + α dz2 dz 4   2 α 3=2 wðzÞ = z C1 z 3

(2:13:107)

d2 wðzÞ dwðzÞ −z + ðα2 z3 + 1 − β2 ÞwðzÞ = 0 dz2 dz   2 α 3=2 wðzÞ = z C2 β=3 z 3

(2:13:108)

z2

z2

d2 wðzÞ dwðzÞ −z + ðα2 z3 + β2 ÞwðzÞ = 0 dz2 dz   2 α 3=2 p ffiffiffiffiffiffiffiffi wðzÞ = C2 z 2 3 3 1−β   d2 wðzÞ dwðzÞ 1 2 3 wðzÞ = 0 + 2 z z + z2 + α dz2 dz 4   1 2α 3=2 p ffiffi ffi C0 wðzÞ = z 3 z z2

d2 wðzÞ dwðzÞ +2z + ðα2 z3 − 2ÞwðzÞ = 0 dz2 dz   1 2 α 3=2 wðzÞ = pffiffiffi C1 z 3 z

(2:13:109)

(2:13:110)

z2

d2 wðzÞ dwðzÞ +2z + ðα2 z3 + β2 ÞwðzÞ = 0 2 dz dz   2 α 3=2 wðzÞ = C1 pffiffiffiffiffiffiffiffiffiffi2ffi z 3 3 1−4β ! 2 dwðzÞ 1 − 9 β2 2 d wðzÞ 2 3 z +2z wðzÞ = 0 + αz + 4 dz2 dz   1 2α 3=2 wðzÞ = pffiffiffi Cβ z 3 z

(2:13:111)

z2

d2 wðzÞ dwðzÞ −2z + ðα2 z3 + β2 ÞwðzÞ = 0 dz2 dz   pffiffiffi 2 α 3=2 wðzÞ = z C1 pffiffiffiffiffiffiffiffiffi2 z 9−β 3 3

(2:13:112)

(2:13:113)

z2

(2:13:114)

2.13 Differential Equations Reducible to the Bessel Differential Equations

55

  d2 wðzÞ dwðzÞ 9 2 3 9  2 wðzÞ = 0 − 2 z z + + α 1 − β dz2 dz 4 4   2 α 3=2 wðzÞ = z3=2 Cβ z 3

(2:13:115)

d2 wðzÞ dwðzÞ + ðβ + 1Þ z + α2 z3 wðzÞ = 0 dz2 dz   2α 3=2 − β=2 wðzÞ = z Cβ=3 z 3

(2:13:116)

d2 wðzÞ dwðzÞ + ðβ + 1Þ z + ðα2 z 3 + β2 ÞwðzÞ = 0 dz2 dz   − ν=2 pffiffiffiffiffiffiffiffiffiffiffiffi 2 α 3=2 wðzÞ = z C1 2 z 2 3 3 ν −4β

(2:13:117)

z2

z2

z2

d2 wðzÞ dwðzÞ + ðβ + 1Þ z + ðα2 z 3 − 2 β2 ÞwðzÞ = 0 2 dz dz   2 α 3=2 wðzÞ = z − 1=2 Cβ z 3 ! 2 dwðzÞ β2 2 d wðzÞ 2 3 + ðβ + 1Þ z wðzÞ = 0 z + αz + 4 dz2 dz   2 α 3=2 wðzÞ = z − β=2 C0 z 3 z2

d2 wðzÞ dwðzÞ − ðβ + 1Þ z + ðα2 z 3 + β2 ÞwðzÞ = 0 dz2 dz   2 α 3=2 ffi wðzÞ = zν=2 + 1 C1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 2 3 3 ðν + 2Þ − 4 β

(2:13:118)

(2:13:119)

z2

z2

d2 wðzÞ dwðzÞ +z + 4z4 wðzÞ = 0 ; d z2 dz

wðzÞ = C0 ðz2 Þ

d2 wðzÞ dwðzÞ + ðβ + 1Þ z + α2 z4 wðzÞ = 0 2 dz dz  2 αz wðzÞ = z − β=2 Cβ=4 2   d2 wðzÞ z dwðzÞ 1 2 4 wðzÞ = 0 z2 + z + + α dz2 2 dz 16 α wðzÞ = z1=4 C0 z2 2

(2:13:120)

(2:13:121)

z2

(2:13:122)

(2:13:123)

56

2 Properties of the Bessel and Related Functions

  d2 wðzÞ z dwðzÞ 1 2 2 4 wðzÞ = 0 + z − 4β + + α dz2 2 dz 16 α wðzÞ = z1=4 Cβ z2 2 z2

z2

d2 wðzÞ dwðzÞ +z + 4ðz4 − β2 ÞwðzÞ = 0 dz2 dz

(2:13:124)

(2:13:125)

wðzÞ = Cβ ðz Þ 2

d2 wðzÞ dwðzÞ +z + ðα2 z4 − 4β2 ÞwðzÞ = 0 dz2 dz  2 αz wðzÞ = Cβ 2

(2:13:126)

d2 wðzÞ dwðzÞ +z + ½α2 z4 − ð2 n + 1Þ2 wðzÞ = 0 dz2 dz  2 z wðzÞ = Cn + 1=2 2

(2:13:127)

d2 wðzÞ dwðzÞ −z + ðα2 z4 + 1ÞwðzÞ = 0 dz2 dz α wðzÞ = z C0 z2 2

(2:13:128)

d2 wðzÞ dwðzÞ −z + ðα2 z4 + β2 ÞwðzÞ = 0 dz2 dz  2 αz p ffiffiffiffiffiffiffiffi wðzÞ = z C1 2 1 − β 2 2

(2:13:129)

z2

z2

z2

z2

d2 wðzÞ dwðzÞ −z + ðα2 z4 − β2 + 1ÞwðzÞ = 0 dz2 dz α wðzÞ = z Cβ z2 2   2 dwðzÞ 1 2 d wðzÞ 2 4 wðzÞ = 0 + 2z z + α z + dz2 dz 4 α 1 wðzÞ = pffiffiffi C0 z2 2 z   d2 wðzÞ dwðzÞ 15 2 4 wðzÞ = 0 + 2z z − z2 + α dz2 dz 4 α 1 wðzÞ = pffiffiffi C1 z2 2 z z2

(2:13:130)

(2:13:131)

(2:13:132)

2.13 Differential Equations Reducible to the Bessel Differential Equations

  d2 wðzÞ dwðzÞ 1 2 2 4 wðzÞ = 0 + 2 z z − 4 β + + α dz2 dz 4  2 1 αz wðzÞ = pffiffiffi Cβ 2 z   2 dwðzÞ 9 2 d wðzÞ 2 4 z wðzÞ = 0 − 2z + α z + dz2 dz 4 α wðzÞ = z3=2 C0 z2 2 ! 2 dwðzÞ 9 − 16β2 2 d wðzÞ 2 4 − 2z wðzÞ = 0 z + α z + 4 dz2 dz α wðzÞ = z3=2 Cβ z2 2 ! 2 dwðzÞ β2 2 d wðzÞ 2 4 z + ðβ + 1Þ z wðzÞ = 0 + αz + 4 dz2 dz

57

z2

(2:13:133)

(2:13:134)

(2:13:135)

(2:13:136)

wðzÞ = z − β=2 C0 ðα z2 Þ d2 wðzÞ dwðzÞ + ðβ + 1Þ z + ðα2 z 4 + λ2 ÞwðzÞ = 0 dz2 dz wðzÞ = z − β=2 C pffiffiffiffiffiffiffiffiffiffiffiffi ðα z2 Þ z2

(2:13:137)

β2 − 4 λ2

1 4

" # d2 wðzÞ dwðzÞ ðβ + 2Þ2 2 4 − ðβ + 1Þ z wðzÞ = 0 z + αz + 4 dz2 dz  2 αz wðzÞ = zβ=2 + 1 C0 2 ! 2 dwðzÞ β2 − 16 2 d wðzÞ 2 4 + ðβ + 1Þ z z + αz + wðzÞ = 0 dz2 dz 4 2

wðzÞ = z z

− β=2

(2:13:138)

(2:13:139)

C1 ðα z Þ 2

d2 wðzÞ dwðzÞ −3 − 9z5 wðzÞ = 0 2 dz dz

(2:13:140)

wðzÞ = z Z2=3 ðz Þ 2

3

d2 wðzÞ dwðzÞ +z + 16z8 wðzÞ = 0 ; wðzÞ = C0 ðz4 Þ d z2 dz  2  α d2 wðzÞ dwðzÞ α 2 wðzÞ = 0 ; wðzÞ = C + z − β z2 + β z2 d z2 dz z z2

(2:13:141) (2:13:142)

58

2 Properties of the Bessel and Related Functions

  d2 wðzÞ z dwðzÞ 1 2 n+2 wðzÞ = 0 + z + + α d z2 3 dz 9   2α n=2 + 1 wðzÞ = z1=3 C0 z n+2

z2

d2 wðzÞ z dwðzÞ + + ðα2 zn + 2 + β2 ÞwðzÞ = 0 d z2 2 dz   2 α n=2 + 1 1=4 p ffiffiffiffiffiffiffiffiffiffiffi ffi wðzÞ = z C1 z 2 2 1 − 16 β =ðn + 2Þ n + 2 ! 2 z dwðzÞ 9 − 4 β2 2 d wðzÞ 2 n+2 z − + wðzÞ = 0 + α z 16 d z2 2 dz   2 α n=2 + 1 3=4 wðzÞ = z Cβ=ðn + 2Þ z n+2

(2:13:143)

z2

z2

d2 wðzÞ dwðzÞ +z + n2 z2n − 2 wðzÞ = 0 d z2 dz

(2:13:144)

(2:13:145)

(2:13:146)

wðzÞ = C0 ðzn Þ d2 wðzÞ dwðzÞ +z + α2 zn + 2 wðzÞ = 0 d z2 dz   2 α n=2 + 1 wðzÞ = C0 z n+2 z2

z2

d2 wðzÞ dwðzÞ +z + ða2 z2 − β2 ÞwðzÞ = 0 d z2 dz

(2:13:147)

(2:13:148)

wðzÞ = Cβ ða zÞ z2

d2 wðzÞ dwðzÞ +z + ½ðn + 2Þ2 zn + 2 − n2 ÞwðzÞ = 0 d z2 dz

wðzÞ = C2 n=ðn + 2Þ ð2 z

n=2 + 1

(2:13:149)

Þ

"

# d2 wðzÞ dwðzÞ ðn + 2Þ2 2 n+2 z +z − wðzÞ = 0 + αz 4 d z2 dz   2 α n=2 + 1 wðzÞ = C2 n=ðn + 2Þ z n+2

(2:13:150)

d2 wðzÞ dwðzÞ +z + ðα2 zn + 2 − β2 ÞwðzÞ = 0 d z2 dz   2 α n=2 + 1 wðzÞ = C2 β=ðn + 2Þ z n+2

(2:13:151)

2

z2

2.13 Differential Equations Reducible to the Bessel Differential Equations

z2

d2 wðzÞ dwðzÞ +z + β2 ðα2 z2 β − 1ÞwðzÞ = 0 d z2 dz

59

(2:13:152)

β

wðzÞ = C1 ðαz Þ z2

d2 wðzÞ dwðzÞ +z − ði z2 + ν2 ÞwðzÞ = 0 d z2 dz

wðzÞ = berν ðzÞ + i beiν ðzÞ wðzÞ = ber−ν ðzÞ + i bei−ν ðzÞ

(2:13:153)

wðzÞ = kerν ðzÞ + i keiν ðzÞ wðzÞ = ker−ν ðzÞ + i kei−ν ðzÞ d2 wðzÞ dwðzÞ −z + α2 zn + 2 wðzÞ = 0 d z2 dz   2 α n=2 + 1 wðzÞ = z C2=ðn + 2Þ z n+2 z2

d2 wðzÞ dwðzÞ −z + ðα2 zn + 2 + 1ÞwðzÞ = 0 d z2 dz   2 α n=2 + 1 wðzÞ = z C0 z n+2   2 dwðzÞ nðn + 4Þ 2 d wðzÞ 2 n+2 wðzÞ = 0 −z − z + α z d z2 dz 4   2 α n=2 + 1 wðzÞ = z C1 z n+2

(2:13:154)

z2

(2:13:155)

(2:13:156)

d2 wðzÞ dwðzÞ −z + ðα2 zn + 2 + β2 ÞwðzÞ = 0 d z2 dz   2α n=2 + 1 wðzÞ = z C pffiffiffiffiffiffiffiffi2 z 2 1 − β =ðn + 2Þ n + 2

(2:13:157)

d2 wðzÞ dwðzÞ + 2z + α2 zn + 2 wðzÞ = 0 dz2 dz   1 2α n=2 + 1 wðzÞ = pffiffiffi C1=ðn + 2Þ z n+2 z

(2:13:158)

d2 wðzÞ dwðzÞ −2z + α2 zn + 2 wðzÞ = 0 d z2 dz   2 α n=2 + 1 5=2 wðzÞ = z C1=ðn + 2Þ z n+2

(2:13:159)

z2

z2

z2

60

2 Properties of the Bessel and Related Functions

d2 wðzÞ dwðzÞ −2z + ðα2 zn + 2 + 2ÞwðzÞ = 0 d z2 dz   2 α n=2 + 1 3=2 wðzÞ = z C1=ðn + 2Þ z n+2

(2:13:160)

d2 wðzÞ dwðzÞ − 2z + ðα2 zn + 2 + β2 ÞwðzÞ = 0 2 dz dz   2α n=2 + 1 wðzÞ = z3=2 Cpffiffiffiffiffiffiffiffiffiffiffi2ffi z 9 − 4 β =ðn + 2Þ n + 2

(2:13:161)

d2 wðzÞ dwðzÞ +βz + α2 z λ + 2 wðzÞ = 0 d z2 dz   2 α ðλ + 2Þ=2 wðzÞ = zð1 − βÞ=2 Cð1 − βÞ=ðλ + 2Þ z λ+2

(2:13:162)

d2 wðzÞ dwðzÞ − βz + ðα2 z n + 2 + βÞwðzÞ = 0 d z2 dz   2 α n=2 + 1 wðzÞ = zðβ + 1Þ=2 Cðβ − 1Þ=ðn + 2Þ z n+2

(2:13:163)

d2 wðzÞ dwðzÞ −βz + ðαzλ + 2 + μ2 ÞwðzÞ = 0 d z2 dz   2 β λ=2 + 1 ðβ + 1Þ=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðzÞ = z C z ðβ + 1Þ2 − 4 μ2 =ðλ + 2Þ λ + 2

(2:13:164)

z2

z2

z2

z2

z2

z2

d2 wðzÞ dwðzÞ − 2 βz + α2 z2 wðzÞ = 0 d z2 dz

Cβ + 1=2 ðα zÞ "  # 2 z dwðzÞ β−1 2 2 d wðzÞ 2 n+2 z + + wðzÞ = 0 + αz d z2 β dz 2β   2 α n=2 + 1 wðzÞ = zðβ − 1Þ=2 β C0 z n+2 wðzÞ = z

(2:13:165)

β + 1=2

d2 wðzÞ dwðzÞ + ðβ + 1Þz + α2 z n + 2 wðzÞ = 0 d z2 dz   2 α n=2 + 1 wðzÞ = z − β=2 Cβ=ðn + 2Þ z n+2

(2:13:166)

z2

(2:13:167)

2.13 Differential Equations Reducible to the Bessel Differential Equations

d2 wðzÞ dwðzÞ + ðβ + 1Þz + α2 z β wðzÞ = 0 d z2 dz   2α β=2 − β=2 wðzÞ = z C1 z β

61

z2

z2

d2 wðzÞ dwðzÞ + ðβ + 1Þz + β2 z β wðzÞ = 0 2 dz dz

wðzÞ = z

− β=2

C1 ð2 z

β=2

(2:13:168)

(2:13:169)

Þ

! d2 wðzÞ dwðzÞ β2 2 n+2 z + ðβ + 1Þz + wðzÞ = 0 + αz 4 d z2 dz   2 α n=2 + 1 wðzÞ = z − β=2 C0 z n+2 ! 2 dwðzÞ 3β2 2 d wðzÞ 2 n+2 + ðβ + 1Þz − wðzÞ = 0 z + αz 4 d z2 dz   2 α n=2 + 1 wðzÞ = z − β=2 C2 ν=ðn + 2Þ z n+2 ! 2 dwðzÞ 3 β2 2 d wðzÞ 2 β + ðβ + 1Þz wðzÞ = 0 z + αz − 4 d z2 dz   2α n=2 + 1 − β=2 C2 wðzÞ = z z β 2

d2 wðzÞ dwðzÞ − ðβ + 1Þz + α2 z n + 2 wðzÞ = 0 d z2 dz   2 α n=2 + 1 ðβ + 2Þ=2 wðzÞ = z Cðβ + 2Þ=ðn + 2Þ z n+2

(2:13:170)

(2:13:171)

(2:13:172)

z2

d2 wðzÞ dwðzÞ + ðβ − 1Þz + ðα2 zn + 2 − 2βÞwðzÞ = 0 2 dz dz   2 α n=2 + 1 wðzÞ = z1 − β=2 Cðβ + 2Þ=ðn + 2Þ z n+2 " # 2 dwðzÞ ðβ + 2Þ2 2 d wðzÞ 2 n+2 z − ðβ + 1Þz + wðzÞ = 0 + αz 4 d z2 dz   2α n=2 + 1 wðzÞ = zβ=2 + 1 C0 z n+2

(2:13:173)

z2

(2:13:174)

(2:13:175)

62

2 Properties of the Bessel and Related Functions

d2 wðzÞ dwðzÞ + ð1 + 2βÞ z + ðα2 z2 λ + μ2 ÞwðzÞ = 0 d z2 dz α wðzÞ = z − β C1pffiffiffiffiffiffiffiffiffiffi zλ β2 − μ2 λ λ

z2

z2

(2:13:176)

d2 wðzÞ dwðzÞ + ð1 − 2 βÞ z + β2 ðz2 β + 1 − β2 ÞwðzÞ = 0 d z2 dz β

(2:13:177)

β

wðzÞ = z Cβ ðz Þ z2

d2 wðzÞ dwðzÞ + ð1 − 2 βÞ z + ðα2 z2 + β2 − λ2 ÞwðzÞ = 0 d z2 dz

(2:13:178)

β

wðzÞ = z Cλ ðα zÞ z2

d2 wðzÞ dwðzÞ + ð1 − 2 βÞ z + ðα2 λ2 zλ + β2 − λ2 μ2 ÞwðzÞ = 0 d z2 dz β

(2:13:179)

λ

wðzÞ = z Cμ ðα z Þ z2

d2 wðzÞ dwðzÞ − ð2βλ − 1Þ z + α2 β2 z2β wðzÞ = 0 2 dz dz βλ

(2:13:180)

β

wðzÞ = z Cλ ðαz Þ z2

d2 wðzÞ dwðzÞ + ð2α − 2β μ + 1Þz + ½β2 λ2 z2β + αðα − 2 β μÞwðzÞ = 0 d z2 dz

;

(2:13:181)

wðzÞ = zβ μ − α Cμ ðλzβ Þ   d2 wðzÞ 1 − α2 + z2 2 dwðzÞ 2 β wðzÞ = 0 − z z + α + 4 d z2 dz   pffiffiffi z=2 2 α β=2 wðzÞ = z e Cα=β z β  2   2 α 2 2 2 d wðzÞ 2 dwðzÞ 2 z + αz + β z + λ wðzÞ = 0 + 4 d z2 dz pffiffiffi wðzÞ = z e − α z=2 C1pffiffiffiffiffiffiffiffiffiffi2 ðβzÞ z2

2

(2:13:182)

(2:13:183)

1−4λ

! d2 wðzÞ 1 − β2 2 2 dwðzÞ 2 2 − 2αz wðzÞ = 0 + α z +λ z+ z 4 d z2 dz pffiffiffi pffiffiffi wðzÞ = z eα z Cβ ð2 λ zÞ 2

(2:13:184)

2.13 Differential Equations Reducible to the Bessel Differential Equations

 2 2  d2 wðzÞ αz 1 2 n+2 2 dwðzÞ wðzÞ = 0 + αz z + + β + 4 d z2 dz 4   pffiffiffi 2β n=2 + 1 wðzÞ = z e − α z=2 C0 z n+2  2 2  2 αz 2 n+2 2 2 d wðzÞ 2 dwðzÞ z + αz + λ wðzÞ = 0 +β z + 4 d z2 dz   pffiffiffi 2β n=2 + 1 wðzÞ = z e − α z=2 Cpffiffiffiffiffiffiffiffiffiffi2 z 1 − 4 λ =ðn + 2Þ n + 2  2 2  d2 wðzÞ αz 2 2 dwðzÞ z2 + αz z wðzÞ = 0 + β + 4 d z2 dz pffiffiffi pffiffiffi wðzÞ = z e − α z=2 C1 ð2 β zÞ  2 2  2 αz 2 2 2 d wðzÞ 2 dwðzÞ z + αz + β z + λ wðzÞ = 0 + 4 d z2 dz pffiffiffi pffiffiffi wðzÞ = z e − α z=2 Cpffiffiffiffiffiffiffiffiffiffi2 ð2 β zÞ

63

z2

(2:13:185)

(2:13:186)

(2:13:187)

(2:13:188)

1−4λ

 2 2  d2 wðzÞ αz 2 3 2 dwðzÞ z + αz + β z − 2 wðzÞ = 0 + 4 d z2 dz   pffiffiffi 2β 3=2 wðzÞ = z e − α z=2 C1 z 3  2  d2 wðzÞ α 2 2 2 dwðzÞ z2 z2 wðzÞ = 0 + αz z + β + 4 d z2 dz   pffiffiffi β 2 wðzÞ = z e − α z=2 C1=4 z 2  2 2  d2 wðzÞ αz 2 3 2 2 dwðzÞ z2 wðzÞ = 0 + αz z + λ + β + 4 d z2 dz   pffiffiffi − α z=2 2 β 3=2 p ffiffiffiffiffiffiffiffiffiffi wðzÞ = z e C1 z 2 3 2 1−4λ ! 2 1 − β2 + α2 z2 2 2 2 d wðzÞ 2 dwðzÞ z + αz wðzÞ = 0 + λz + 4 d z2 dz pffiffiffi pffiffiffi wðzÞ = z e − α z=2 Cβ ð2 λ zÞ 2

(2:13:189)

(2:13:190)

(2:13:191)

(2:13:192)

64

2 Properties of the Bessel and Related Functions

! d2 wðzÞ 1 − 9 β2 + α2 z2 2 3 2 dwðzÞ z + αz wðzÞ = 0 + λz + 4 d z2 dz   pffiffiffi − α z=2 2 λ 3=2 Cβ wðzÞ = z e z 3 ! 2 1 − 16β2 + α2 z2 2 4 2 d wðzÞ 2 dwðzÞ + αz wðzÞ = 0 z + λz + 4 d z2 dz   pffiffiffi − α z=2 λ 2 Cβ wðzÞ = z e z 2 2

d2 wðzÞ dwðzÞ + z ð1 + 2α zÞ + ðα z + β2 − λ2 ÞwðzÞ = 0 2 dz dz qffiffiffiffiffiffiffiffiffiffiffiffiffi wðzÞ = e − α z Cλ ð β2 − α2 zÞ

(2:13:193)

(2:13:194)

z2

d2 wðzÞ dwðzÞ + z ðβ + 1 − 2α zÞ + z ½α ðβ + 1Þ− 1 − α2 z wðzÞ = 0 d z2 dz pffiffiffi wðzÞ = z − β=2 eα z Cβ ð2 zÞ   d2 wðzÞ 1 2 2 2 2 2β z2 wðzÞ = αμ + 1 β2 zβ μ + β + 1=2 + α β z + λ − β d z2 4

z2

wðzÞ = z

1=2

wðzÞ = z

− ðα − 1Þ=2

β

β

(2:13:196)

(2:13:197)

β

½A1 Jλ ðα z Þ + A2 Y λ ðα z Þ + sμ, λ ðαz Þ " # 2 dwðzÞ ðα − 1Þ2 2 2 d wðzÞ 2 z + αz wðzÞ = β zμ − ðα + 1Þ=2 + z −λ + 4 d z2 dz

z2

(2:13:195)

(2:13:198)

½A1 Jλ ðzÞ + A2 Yλ ðzÞ + β sμ, λ ðzÞ

d2 wðzÞ dwðzÞ + ð1 − 2αÞ z + ½β2 z2 + α2 − λ2 wðzÞ = βμ + 1 zα + μ − 1 2 dz dz

(2:13:199)

α

wðzÞ = z ½A1 Jλ ðβzÞ + A2 Yλ ðβzÞ + sμ, λ ðβzÞ z2

d2 wðzÞ dwðzÞ + ð1 − 2αÞ z + ½β2 λ2 z2λ + α2 − λ2 ν2 wðzÞ = βμ + 1 λ2 zα + λ μ + λ d z2 dz α

λ

λ

(2:13:200)

λ

wðzÞ = z ½A1 Jν ðβz Þ + A2 Y ν ðβ z Þ + sμ, ν ðβ z Þ z2

d2 wðzÞ dwðzÞ + ð1 − 2αλÞz + α2 β2 z2 α wðzÞ = α2 βμ + 1 zα ðλ + μ + 1Þ 2 dz dz αλ

α

α

α

wðzÞ = z ½A1 Jλ ðβ z Þ + A2 Yλ ðβ z Þ + sμ, λ ðβ z Þ

(2:13:201)

2.13 Differential Equations Reducible to the Bessel Differential Equations

d3 wðzÞ dwðzÞ − 2z 2wðzÞ = 0 d z3 dz

65

(2:13:202)

wðzÞ = A1 Ai ðzÞ + A2 AiðzÞBiðzÞ + A3 A1 Bi ðzÞ 2

2

d3 wðzÞ dwðzÞ zν + z2 α − 2 + ðα − 1Þz2 α − 3 wðzÞ = 0 ; ζ = 3 2α dz dz n  2  2 o wðzÞ = z A1 J1=2 α ðζ Þ + A2 J1=2 α ðζ ÞYν ðζ Þ + A1 Y1=2 α ðζ Þ z

d3 wðzÞ d2 wðzÞ dwðzÞ − ðz + 2 αÞ + ðz − 2 α − 1Þ + ðz − 1ÞwðzÞ = 0 3 2 dz dz dz

wðzÞ = A1 e + z z

α+1

(2:13:203)

(2:13:204)

½A2 Iα + 1 ðzÞ + A3 Kα + 1 ðzÞ

d3 wðzÞ d2 wðzÞ dwðzÞ − 4ðz + α − 1Þ + ð2z + 6α − 5Þ + ð1 − 2 αÞwðzÞ = 0 3 dz dz2 dz      iz iz + A3 Y α wðzÞ = A1 ez + zα ez=2 A2 Jα 2 2

2z

d3 wðzÞ d2 wðzÞ dwðzÞ + 3 z + ð1 + z2 Þ =0 dz3 d z2 dz

z2

(2:13:205)

(2:13:206)

wðzÞ = A1 + A2 Ji0 ðzÞ + A3 Yi0 ðzÞ d3 wðzÞ d2 wðzÞ dwðzÞ + 3z + ½4z2 − 4n2 + 1 + 4z wðzÞ = 0 3 2 dz dz dz

z2

(2:13:207)

wðzÞ = A1 Jn2 ðzÞ + A2 Jn ðzÞYn ðzÞ + A3 Yn2 ðzÞ z2

d3 wðzÞ d2 wðzÞ dwðzÞ + 3z + ½4α2 z2 α − 4α2 β2 + 1 + 4α2 z2 α − 1 wðzÞ = 0 3 dz d z2 dz

(2:13:208)

wðzÞ = A1 Jβ2 ðzα Þ + A2 Jβ ðzα ÞYβ ðzα Þ + A3 Yβ2 ðzα Þ   d3 wðzÞ d2 wðzÞ 1 dwðzÞ 2 2 + 2z ð1 − zÞ + z − 2z − α + d z3 d z2 4 dz   1 wðzÞ = 0 + α2 − 4      pffiffiffi iz iz + A3 Yα i wðzÞ = A1 ez + z ez=2 A2 Jα 2 2 z2

(2:13:209)

66

2 Properties of the Bessel and Related Functions

  d3 wðzÞ d2 wðzÞ 1 dwðzÞ 2 2 − 2z ð1 − zÞ + z − 2z − α + d z3 d z2 4 dz   1 wðzÞ = 0 + α2 − 4 h z  z i pffiffiffi wðzÞ = A1 ez + z ez=2 A2 Iα + A3 K α 2 2 z2

(2:13:210)

d3 wðzÞ d2 wðzÞ dwðzÞ − z ðz + αÞ + α ð1 + 2zÞ − αð1 + zÞwðzÞ = 0 3 2 dz dz dz pffiffiffiffiffiffi pffiffiffiffiffiffi wðzÞ = A1 ez + zðα + 1Þ=2 ½A2 Jα + 1 ð2 α zÞ + A3 Yα + 1 ð2 αzÞ   3 d2 wðzÞ 1 dwðzÞ 2 d wðzÞ 2 2 z − z ðz − 2Þ − z +α − d z3 d z2 4 dz   1 wðzÞ = 0 + z2 − 2 z + α2 − 4 pffiffiffi wðzÞ = A1 ez + z ½A2 Iα ðzÞ + A3 Kα ðzÞ z2

z3

d3 wðzÞ d2 wðzÞ + ð4z3 + α zÞ − αwðzÞ = 0 3 dz d z2

(2:13:211)

(2:13:212)

(2:13:213)

wðzÞ = A1 Jα2 ðzÞ + A2 Jα ðzÞ Yα ðzÞ + A3 Yα2 ðzÞ z3

d3 wðzÞ dwðzÞ + ½4 z3 + ð1 − 4 α2 Þz  − ð1 − 4 α2 ÞwðzÞ = 0 d z3 dz

(2:13:214)

wðzÞ = z ½A1 Jα2 ðzÞ + A2 Jα ðzÞYα ðzÞ + A3 Yα2 ðzÞ z

d4 wðzÞ d3 wðzÞ − + 4 z3 wðzÞ = 0 dz4 dz3

wðzÞ = ½A1 AiðzÞ Aið− zÞ + A2 AiðzÞ Bið− zÞ + A3 Aið− zÞ BiðzÞ + A4 BiðzÞ Bið− zÞ (2:13:215) d4 wðzÞ d3 wðzÞ dwðzÞ − 10z − 10z + 9 z2 wðzÞ = 0 d z4 d z3 dz 3

2

(2:13:216) 2

wðzÞ = A1 ½AiðzÞ + A2 ½AiðzÞ  BiðzÞ + A3 AiðzÞ½BiðzÞ  + A4 ½BiðzÞ

3

d4 wðzÞ d3 wðzÞ d2 wðzÞ + 6z +6 − α2 wðzÞ = 0 4 3 dz dz d z2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 1 wðzÞ = pffiffiffi ½A1 J1 ð2 αzÞ + A2 Y1 ð2 α zÞ + A3 J1 ð2i αzÞ + A4 Y1 ð2i αzÞ z z2

(2:13:217)

2.13 Differential Equations Reducible to the Bessel Differential Equations

67

d4 wðzÞ d3 wðzÞ d2 wðzÞ + 8z + 12 − α2 wðzÞ = 0 dz4 dz3 dz2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 1 wðzÞ = ½A1 J1 ð2 α zÞ + A2 Y1 ð2 α zÞ + A3 J1 ð2i α zÞ + A4 Y1 ð2i α zÞ z

(2:13:218)

d4 wðzÞ d3 wðzÞ d2 wðzÞ + 2 ðα + 2Þz + ðα + 1Þðα + 2Þ − β4 wðzÞ = 0 4 3 dz dz dz2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi wðzÞ = z − α=2 ½A1 Jα ð2 βzÞ + A2 Yα ð2 β zÞ + A3 Iα ð2 βzÞ + A4 Kα ð2 βzÞ

(2:13:219)

z2

z2

z3

3 d4 wðzÞ d2 wðzÞ d wðzÞ 2 d wðzÞ + 2z − z + − α4 z3 wðzÞ = 0 dz4 d z3 d z2 dz

(2:13:220)

wðzÞ = A1 J0 ðαzÞ + A2 Y0 ðα zÞ + A3 I0 ð α zÞ + A4 K0 ðαzÞ d4 wðzÞ d3 wðzÞ d2 wðzÞ + 6 z2 + 6 z2 + 4α4 z2 wðzÞ = 0 4 3 dz dz dz2 pffiffiffi pffiffiffi 1 wðzÞ = pffiffiffi ½A1 ber1 ð23=2 α zÞ + A2 bei1 ð23=2 α zÞ z pffiffiffi pffiffiffi + A3 ker1 ð23=2 α zÞ + A4 kei1 ð23=2 α zÞ z4

d4 wðzÞ d3 wðzÞ d2 wðzÞ + 2 ð2 − nÞ z3 + ð1 − nÞð2 − nÞz2 4 3 dz dz dz2     2 α n=2 2α n=2 + A2 Y1=n − α4 z2 n wðzÞ = 0 ; wðzÞ = A1 J1=n z z n n     2α n=2 2 α n=2 + A4 K1=n + A3 I1=n z z n n   3 2 d4 wðzÞ dwðzÞ 3 d wðzÞ 2 2 d wðzÞ z4 + 2z − ð1 + 2α Þ z − z dz4 dz3 d z2 dz

(2:13:221)

z4

+ ðβ4 α4 − α2 + z4 ÞwðzÞ = 0

(2:13:222)

(2:13:223)

wðzÞ = A1 Jα ðβ zÞ + A2 Yα ðβ zÞ + A3 Iα ðβzÞ + A4 Kα ðβzÞ wðzÞ = ber ± α ðβ zÞ z4

;

bei ± α ðβzÞ;

ker ± α ðβ zÞ;

kei ± α ðβ zÞ

d4 wðzÞ d3 wðzÞ d2 wðzÞ + 6z3 + ½4z4 + ð7 − α2 − β2 Þz2  4 3 dz dz dz2 + zð16z2 + 1 − α2 − β2 Þ

dwðzÞ + ð8z2 + α2 β2 ÞwðzÞ = 0 dz

wðzÞ = A1 Jðα + βÞ=2 ðzÞJðα − βÞ=2 ðzÞ + A2 Yðα + βÞ=2 ðzÞJðα − βÞ=2 ðzÞ + A3 Jðα + βÞ=2 ðzÞ Yðα − βÞ=2 ðzÞ + A4 Yðα + βÞ=2 ðzÞYðα − βÞ=2 ðzÞ

(2:13:224)

68

2 Properties of the Bessel and Related Functions

d2 wðzÞ + e2 z wðzÞ = 0 ; dz2

wðzÞ = C0 ðez Þ

d2 wðzÞ + α2 eα z wðzÞ = 0 dz2

;

wðzÞ = C0 ð2 eα z=2 Þ

d2 wðzÞ + α2 eβ z wðzÞ = 0 dz2

;

wðzÞ = C0

  2 α β z=2 e β

d2 wðzÞ + α2 β2 eα z wðzÞ = 0 ; wðzÞ = C0 ð2 βeα z=2 Þ dz2     d2 wðzÞ 1 α βz 2 βz wðzÞ = 0 ; wðzÞ = C + α e − 2 βe β dz2 4 β

(2:13:225) (2:13:226) (2:13:227)

(2:13:228) (2:13:229)

d2 wðzÞ + ðα2 eα z − 4ÞwðzÞ = 0 dz2

;

wðzÞ = C4=α ð2 eα z=2 Þ

(2:13:230)

d2 wðzÞ + ð4e2 z − α2 ÞwðzÞ = 0 dz2

;

wðzÞ = Cα ð2 ez Þ

(2:13:231)

d2 wðzÞ + ð4β2 e2 z − α2 ÞwðzÞ = 0 dz2

;

wðzÞ = Cα ð2 βez Þ

(2:13:232)

d2 wðzÞ + α2 ð4e2 α z − 1ÞwðzÞ = 0 dz2

;

wðzÞ = C1 ð2 eα z Þ

(2:13:233)

d2 wðzÞ + α2 ð4e2 α z − α2 ÞwðzÞ = 0 dz2

wðzÞ = Cα ð2 eα z Þ

(2:13:234)

wðzÞ = C2 β ð2α ez Þ

(2:13:235)

;

d2 wðzÞ + ðα2 e2 z − β2 Þ wðzÞ = 0 ; dz2

  α βz ; wðzÞ = C1 e β   d2 wðzÞ α βz 4 2 2βz + ðα e − β ÞwðzÞ = 0 ; wðzÞ = C e β dz2 β d2 wðzÞ + ðα2 e2 β z − β2 ÞwðzÞ = 0 dz2

d2 wðzÞ n−1 + ðα2 e2 β z − β2 n ÞwðzÞ = 0 dz2 ! α βn − 1 z wðzÞ = Cβ n − 1 e β   d2 wðzÞ 1 2 β=z wðzÞ = 0 + α e − dz2 4

;

(2:13:236)

(2:13:237)

(2:13:238)

  α wðzÞ = C1=β 2 eβ=2 z β

(2:13:239)

2.13 Differential Equations Reducible to the Bessel Differential Equations

69

z4

d2 wðzÞ + α2 eα=z wðzÞ = 0 dz2

;

wðzÞ = z C0 ðeα=2 z Þ

(2:13:240)

z4

d2 wðzÞ + α2 eα=z wðzÞ = 0 dz2

;

wðzÞ = z C0 ðeα=2 z Þ

(2:13:241)

z4

d2 wðzÞ + ðe2=z − α2 ÞwðzÞ = 0 dz2

;

wðzÞ = z Cα ðe1=z Þ

d2 wðzÞ + ðα2 eβ=z − λ2 ÞwðzÞ = 0 dz2   α λ=z wðzÞ = z C2 λ=β 2 e β

(2:13:242)

z4

d2 wðzÞ dwðzÞ + αβ + α2 λ2 eα z wðzÞ = 0 2 dz dz − α β z=2

(2:13:244)

α z=2

Cβ ð2 λe Þ   d2 wðzÞ dwðzÞ 1 2 βz wðzÞ = 0 + e + + α dz2 dz 4   2 α β z=2 wðzÞ = e − α z=2 C0 e β wðzÞ = e

(2:13:243)

d2 wðzÞ dwðzÞ +2 + ðα2 eβ z + 1Þ wðzÞ = 0 dz2 dz   2 α β z=2 wðzÞ = e − z C0 e β d2 wðzÞ dwðzÞ −2 + ðα2 eβ z + 1Þ wðzÞ = 0 dz2 dz   2 α β z=2 wðzÞ = ez C0 e β   d2 wðzÞ dwðzÞ α2 2 λz wðzÞ = 0 +α + βe + 4 dz2 dz   2 β λ z=2 wðzÞ = e − α z=2 C0 e λ   d2 wðzÞ dwðzÞ α2 − 4μ2 2 λz wðzÞ = 0 + α e + + β 4 dz2 dz   2 β λz=2 wðzÞ = e − α z=2 C2 μ=λ e λ

(2:13:245)

(2:13:246)

(2:13:247)

(2:13:248)

(2:13:249)

70

2 Properties of the Bessel and Related Functions

d2 wðzÞ dwðzÞ α2 − 4λ2 +α + β2 e2 z + 2 4 dz dz

! wðzÞ = 0

(2:13:250)

wðzÞ = e − α z=2 Cλ ðβe z Þ d2 wðzÞ dwðzÞ +α + ðβ eλ z + δÞwðzÞ = 0 dz2 dz ! pffiffiffi − α z=2 pffiffiffiffiffiffiffiffiffiffi 2 β λ z=2 wðzÞ = e C α2 − 4δ z λ λ d2 wðzÞ dwðzÞ + α ðβ − 2Þ + ½α2 ð1 − βÞ + β2 λ2 eβ z wðzÞ = 0 2 dz dz ðα − β=2Þ z

(2:13:252)

β z=2

Cα ð2λe Þ   d2 wðzÞ z dwðzÞ 3 2 2 2 βz z2 wðzÞ = 0 + ðα e − λ Þ− + z d z2 2 dz 16   2 α β z=2 wðzÞ = z − 1=4 C2 λ=β e β wðzÞ = e

(2:13:251)

  d2 wðzÞ z dwðzÞ 1 2 β= z 2 3 wðzÞ = 0 + ðα e − λ Þ− + d z2 2 dz z2 16   2α β=2 z 3=4 wðzÞ = z C2 λ=β e β   d2 wðzÞ z dwðzÞ 1 2 β= z 2 5 z2 wðzÞ = 0 + − ðα e − λ Þ + + d z2 2 dz z2 16   2 α β=2 z wðzÞ = z5=4 C2 λ=β e β   2 z dwðzÞ 5 2 2 d wðzÞ 2 2 βz z wðzÞ = 0 − + z ðα e − λ Þ− d z2 2 dz 16   2α β z=2 wðzÞ = z1=4 C2 λ=β e β

(2:13:253)

z2

d2 wðzÞ dwðzÞ 1 +z + ðα2 z2 ez − Þ wðzÞ = 0 d z2 dz 4 1 wðzÞ = pffiffiffi C0 ð2αez=2 Þ z

(2:13:254)

(2:13:255)

(2:13:256)

z2

(2:13:257)

2.13 Differential Equations Reducible to the Bessel Differential Equations

! # " d2 wðzÞ dwðzÞ β2 1 2 2 z z +z − + z ðα e − wðzÞ = 0 4 d z2 dz 4

71

2

(2:13:258)

1 wðzÞ = pffiffiffi Cβ ð2 αez=2 Þ z ! # " d2 wðzÞ dwðzÞ β2 1 2 2 αz z +z − + z α e − wðzÞ = 0 4 d z2 dz 4 2

1 wðzÞ = pffiffiffi C2 β=α ð2 α eα z=2 Þ z   d2 wðzÞ dwðzÞ 1 2 2 2 2z z2 wðzÞ = 0 + z ðα e − β Þ− + z d z2 dz 4 1 wðzÞ = pffiffiffi Cβ ðαez=2 Þ z   d2 wðzÞ dwðzÞ 1 2 2 2 βz wðzÞ = 0 + z ðα e − λ Þ− + z d z2 dz 4   1 2 α β z=2 wðzÞ = pffiffiffi C2 λ=β e β z   2 dwðzÞ 1 2 2 d wðzÞ 2 2 2βz z wðzÞ = 0 +z + z ðα e − λ Þ− d z2 dz 4   1 α βz wðzÞ = pffiffiffi Cλ=β e β z  2  d2 wðzÞ dwðzÞ α 2 =z 1 z2 wðzÞ = 0 + z e − + z2 d z2 dz 4 pffiffiffi wðzÞ = z C0 ð αe1= z Þ   2 dwðzÞ 1 2 2= z 2 1 2 d wðzÞ z wðzÞ = 0 +z + 2 ðα e − β Þ− d z2 dz z 4 pffiffiffi wðzÞ = z Cβ ðαe1=z Þ ! # " 2 dwðzÞ 1 β2 1 2 d wðzÞ 2 1= z z +z − + 2 α e − wðzÞ = 0 4 d z2 dz z 4 pffiffiffi wðzÞ = z Cβ ð2αe1=2 z Þ

(2:13:259)

(2:13:260)

z2

(2:13:261)

(2:13:262)

(2:13:263)

(2:13:264)

(2:13:265)

72

2 Properties of the Bessel and Related Functions

! # " d2 wðzÞ dwðzÞ 1 λ2 1 2 β= z z +z − wðzÞ = 0 + 2 α e − 4 d z2 dz z 4   pffiffiffi 2 α β=2 z wðzÞ = z Cλ=β e β   d2 wðzÞ dwðzÞ 3 2 2 2z wðzÞ = 0 − z z e + z2 + α d z2 dz 4 pffiffiffi wðzÞ = z C0 ðα e z Þ   d2 wðzÞ dwðzÞ 3 2 2 2 2z wðzÞ = 0 − z ðα e − β Þ + z2 + z d z2 dz 4 pffiffiffi wðzÞ = z Cβ ðαe z Þ ! " # 2 2 d wðzÞ dwðzÞ λ 3 z2 −z + + z2 α2 eβ z − wðzÞ = 0 4 d z2 dz 4   pffiffiffi 2 α β z=2 wðzÞ = z Cλ=β e β  2  d2 wðzÞ dwðzÞ α 2= z 3 wðzÞ = 0 − z e + z2 + z2 d z2 dz 4 2

wðzÞ = z

3=2

C0 ðα e

1=z

" d2 wðzÞ dwðzÞ 1 z −z + 2 d z2 dz z

α e

2 1= z

β2 − 4

!

# 3 + wðzÞ = 0 4

wðzÞ = z3=2 Cβ ð2αe 1=2 z Þ   2 dwðzÞ 1 2 2= z 2 3 2 d wðzÞ −z z + 2 ðα e − β Þ + wðzÞ = 0 d z2 dz z 4 wðzÞ = z

Cβ ð2αe

(2:13:267)

(2:13:268)

(2:13:269)

(2:13:270)

Þ

2

3=2

(2:13:266)

1= z

(2:13:271)

(2:13:272)

Þ

! " # d2 wðzÞ dwðzÞ 1 λ2 3 2 β= z z −z + + 2 α e − wðzÞ = 0 4 d z2 dz z 4   2α β=2 z 3=2 wðzÞ = z Cλ=β e β

(2:13:273)

d2 wðzÞ dwðzÞ + 2z + α2 ðe2 α z − α2 Þz2 wðzÞ = 0 d z2 dz 1 wðzÞ = Cα ðe αz Þ z

(2:13:274)

2

z2

2.13 Differential Equations Reducible to the Bessel Differential Equations

73

d2 wðzÞ dwðzÞ + 2z + ðα2 e2 z − β2 Þz2 wðzÞ = 0 d z2 dz 1 wðzÞ = Cβ ðαe z Þ z

(2:13:275)

d2 wðzÞ dwðzÞ + 2z + ðα2 e β z − λ2 Þz2 wðzÞ = 0 d z2 dz   1 2 α βz=2 wðzÞ = C2 λ=β e z β

(2:13:276)

d2 wðzÞ dwðzÞ + 2z + ðα2 e2 β z − λ2 Þ z2 wðzÞ = 0 d z2 dz   1 α βz wðzÞ = Cλ=β e z β

(2:13:277)

d2 wðzÞ dwðzÞ + 2z + ðα2 e2 β z − β2 λ2 Þ z2 wðzÞ = 0 d z2 dz   1 α βz wðzÞ = Cλ e z β

(2:13:278)

z2

z2

z2

z2

z2

d2 wðzÞ dwðzÞ α2 α= z 2 + 2 z + 2 ðe − β ÞwðzÞ = 0 z d z2 dz

wðzÞ = C2 β ðα e z2

α=2 z

Þ

d2 wðzÞ dwðzÞ 1 + 2z + 2 ðα2 e2= z − β2 Þ wðzÞ = 0 d z2 dz z

wðzÞ = Cβ ðα e

1=z

(2:13:279)

(2:13:280)

Þ

d2 wðzÞ dwðzÞ 1 + 2z + 2 ðα2 eβ= z − λ2 Þ wðzÞ = 0 2 dz dz z   2α β=2 z wðzÞ = C2 λ=β e β

z2

z2

d2 wðzÞ dwðzÞ − 2z + ½z2 ðα2 e2 z − β2 Þ + 2 wðzÞ = 0 2 dz dz

(2:13:281)

(2:13:282)

wðzÞ = z Cβ ðα e Þ z

d2 wðzÞ dwðzÞ − 2z + ½z2 ðα2 eβ z − λ2 Þ + 2 wðzÞ = 0 d z2 dz   2α β z=2 wðzÞ = z C2 λ=β e β

z2

(2:13:283)

74

2 Properties of the Bessel and Related Functions

z2

  d2 wðzÞ dwðzÞ 1 2 2= z 2 − 2 z ðα e − β Þ + 2 wðzÞ = 0 + d z2 dz z2

wðzÞ = z Cβ ðαe 2

1=z

(2:13:284)

Þ

  d2 wðzÞ dwðzÞ 1 2 β= z 2 − 2 z ðα e − λ Þ + 2 wðzÞ = 0 + d z2 dz z2   2 α β=2 z 2 wðzÞ = z C2 λ=β e β   2 dwðzÞ 1 2 2= z 2 2 d wðzÞ z − 2z + 2 ðα e − β Þ + 2 wðzÞ = 0 d z2 dz z z2

wðzÞ = z Cβ ðαe 2

1=z

(2:13:285)

(2:13:286)

Þ

  d2 wðzÞ dwðzÞ 1 2 β= z 2 − 2 z ðα e − λ Þ + 2 wðzÞ = 0 + d z2 dz z2   2α β=2 z 2 wðzÞ = z C2 λ=β e β z2

z2

d2 wðzÞ dwðzÞ + 2α z + ½ðβ2 e2 μ z − λ2 Þμ2 z2 + αðα − 1Þ wðzÞ = 0 d z2 dz

(2:13:287)

(2:13:288)

wðzÞ = z − α Cλ ðβeμ z Þ z2

d2 wðzÞ dwðzÞ − 2α z + ½αðα + 1Þ− z2 ð1 − 4β2 λ2 e2 β z Þ wðzÞ = 0 2 dz dz α

(2:13:289)

βz

wðzÞ = z C1=β ð2λe Þ d2 wðzÞ dwðzÞ β2 2 2 z z + ðβ + 1Þz z e − + α 4 d z2 dz 2

wðzÞ = z

− ðβ + 1Þ

C0 ð2αe

z=2

! wðzÞ = 0

(2:13:290)

Þ

! d2 wðzÞ dwðzÞ β2 − 1 2 2 βz + ðβ + 1Þz + αz e + wðzÞ = 0 z d z2 dz 4   2 α β z=2 wðzÞ = z − ðβ + 1Þ=2 C0 e β " # 2 dwðzÞ β2 − 1 2 2 d wðzÞ 2 2z + ðβ + 1Þz z + z ð4e − β Þ + wðzÞ = 0 d z2 dz 4 2

wðzÞ = z − ðβ + 1Þ=2 Cβ ð2 e z Þ

(2:13:291)

(2:13:292)

2.13 Differential Equations Reducible to the Bessel Differential Equations

! " # d2 wðzÞ dwðzÞ β2 β2 − 1 2 2 λz z + ðβ + 1Þz + + z α e − wðzÞ = 0 4 d z2 dz 4   2 α λ z=2 − ðβ + 1Þ=2 Cβ=λ wðzÞ = z e β " # 2 dwðzÞ β2 − 1 2 d wðzÞ 2 2 2λ z 2 + ðβ + 1Þz z + z ðα e − μ Þ + wðzÞ = 0 d z2 dz 4 α eλz wðzÞ = z − ðβ + 1Þ=2 Cμ=λ λ ! 2 dwðzÞ e1= z β2 − 1 2 d wðzÞ z + ðβ + 1Þz + + wðzÞ = 0 z2 d z2 dz 4

75

2

wðzÞ = z

ð1 − βÞ=2

C0 ð2e

1=2 z

(2:13:290)

(2:13:291)

(2:13:292)

Þ

" # d2 wðzÞ dwðzÞ 1 β2 − 1 2 2= z + ðβ + 1Þz + 2 ð4e − β Þ + z d z2 dz z 4

(2:13:293)

; wðzÞ = z ð1 − βÞ=2 Cβ ð2e 1= z Þ " # 2 dwðzÞ 1 2 2= z 2 β2 − 1 2 d wðzÞ + ðβ + 1Þz + 2 ðα e − λ Þ + z d z2 dz z 4

(2:13:294)

2

wðzÞ = 0

ð1 − βÞ=2

Cλ ðα e Þ " # 2 dwðzÞ 1 2 2 λ= z β2 − 1 2 d wðzÞ 2 + ðβ + 1Þz −μ Þ+ + 2 ðα e z d z2 dz z 4 α wðzÞ = 0 ; wðzÞ = z ð1 − βÞ=2 Cμ=λ e λ= z λ wðzÞ = 0 ;

z2

wðzÞ = z

1= z

d2 wðzÞ dwðzÞ + z ðβz − 2 αÞ + ½αðα + 1Þ− α β z + λ2 ez wðzÞ = 0 2 dz dz α

wðzÞ = z e

− β z=2

Cβ ð2 λe

z=2

(2:13:295)

(2:13:296)

Þ

d2 wðzÞ dwðzÞ 2 − ð2 αz2 + 1Þ + 4β2 z3 e2 λ z wðzÞ = 0 d z2 dz ! pffiffiffi β λ z2 α2 z=2 wðzÞ = e Cα=2 λ z λ z

z2

d2 wðzÞ dwðzÞ − 2αz + ½αðα + 1Þ− z2 ð1 − 4β2 λ2 e2 β z Þ wðzÞ = 0 2 dz dz α z

βz

wðzÞ = z e C1=β ð2 λe Þ

(2:13:297)

(2:13:298)

76

2 Properties of the Bessel and Related Functions

z4

d2 wðzÞ dwðzÞ + z5 + ðz2 − α2 ÞwðzÞ = 0 dz2 dz

;

z4

d2 wðzÞ dwðzÞ + z5 − ðz2 + α2 ÞwðzÞ = 0 2 dz dz

;

  1 z   1 wðzÞ = Zα z wðzÞ = Cα

(2:13:299)

(2:13:300)

d2 wðzÞ dwðzÞ +z + α2 ½lnðβ zÞn wðzÞ = 0 d z2 dz α pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðzÞ = lnðβzÞ C1=2 n ½lnðβzÞn n

(2:13:301)

d2 wðzÞ dwðzÞ + ðz − 2z2 tan zÞ − ðz tan z + α2 Þ wðzÞ = 0 d z2 dz 1 wðzÞ = Cα ðzÞ cos z

(2:13:302)

d2 wðzÞ dwðzÞ + ðz + 2z2 cot zÞ + ðz cot z − α2 Þ wðzÞ = 0 d z2 dz 1 wðzÞ = Cα ðzÞ sin z

(2:13:303)

z2

z2

z2

z2

d2 wðzÞ dwðzÞ + z ðα + 1 − 2z tan zÞ − 2 dz dz   z 2  wðzÞ = 0 ðα + 1Þz ðtan z − 1Þ− ðz tan zÞ2 − cos z

wðzÞ = z2

z2

pffiffiffi z − α=2 Cα ð2 zÞ cos z

d2 wðzÞ dwðzÞ + z ðα + 1 + 2z cot zÞ + d z2 dz   z 2  wðzÞ = 0 ðα + 1Þz ðcot z + 1Þ− ðz cot zÞ2 − sin z

wðzÞ =

(2:13:304)

(2:13:305)

pffiffiffi z − α=2 Cα ð2 zÞ sin z

d2 wðzÞ dwðzÞ + z ðα + 1 + 2z tan zÞ 2 dz dz

+ ½ðα + 1Þ z ðtantz + 1Þ− 2ðz tan zÞ2 − z2 wðzÞ = 0 pffiffiffi wðzÞ = cos z Cα ð2 zÞ

(2:13:306)

2.13 Differential Equations Reducible to the Bessel Differential Equations

z2

77

d2 wðzÞ dwðzÞ + z ½ 1 + β λ + 2αz tanðαzÞ + ½α2 z2 ð1 + 2ðtanðα zÞÞ2 d z2 dz + αz ð1 + βλÞ tanðαzÞ + β2 μ2 zλ wðzÞ = 0

(2:13:307)

wðzÞ = z − β λ=2 cosðα zÞ Cβ ð2μ zλ=2 Þ z2

d2 wðzÞ dwðzÞ + z ðα + 1 − 2z cot zÞ d z2 dz

− ½ðα + 1Þz ðcot z − 1Þ− 2 ðz cot zÞ2 − z2 wðzÞ = 0 pffiffiffi wðzÞ = sin z Cα ð2 zÞ z2

(2:13:308)

d2 wðzÞ dwðzÞ + z ½ 1 + β λ + 2αz cotðα zÞ d z2 dz + ½α2 z2 ð1 + 2ðcotðαzÞÞ2 − α z ð1 + βλÞ cotðαzÞ + β2 μ2 zλ  wðzÞ = 0

;

(2:13:309)

wðzÞ = z − β λ=2 sinðα zÞ Cβ ð2 μzλ=2 Þ

d2 wðzÞ dwðzÞ + tan z + α2 ðcos zÞ2 ðsin zÞ2 n − 2 wðzÞ = 0 dz2 dz hα i pffiffiffiffiffiffiffiffiffiffi wðzÞ = sin z C1=2 n ðsin zÞn n

(2:13:310)

3 Differentiation and Integration with Respect to the Order of the Bessel and Related Functions 3.1 First Derivatives of the Bessel and Related Functions with Respect to the Order ν: Unrestricted Values of the Order ν Two solutions of the Bessel differential equation (2.3.1) z2

d2 wðzÞ dwðzÞ +z + ðz2 − ν2 ÞwðzÞ = 0 d z2 dz

(3:1:1)

depend on continuously changing variable z (argument z) and a prescribed fixed value ν which is called the order of Bessel function or sometimes the index of it: wðzÞ = A1 Jν ðzÞ + A2 J− ν ðzÞ

(3:1:2)

where A1 and A2 are the integration constants. These Bessel functions of the first kind in eq. (3.1.2) are linearly independent solutions of the Bessel differential equation for any real or complex value of ν. However, when the order ν is an integer n, it follows from (2.2.10) that J−n ðzÞ = ð− 1Þn Jn ðzÞ

(3:1:3)

and therefore in order to determine the second independent solution of differential equation, the Bessel functions of the second kind are introduced: Yn ðzÞ =

cosðπ nÞJn ðzÞ − J−n ðzÞ sinðπ nÞ

(3:1:4)

If ν = n, the function Yn(z) is undefined, both the numerator and the denominator vanish and the resulting form of 0/0 is resolved by applying L’Hôpital rule to (3.1.4)   cosðπ nÞJn ðzÞ − J− n ðzÞ Yn ðzÞ = lim Yν ðzÞ = lim ν!n ν!n sinðπ nÞ Yn ðzÞ =

1 ∂ ½Jν ðzÞ−ð− 1Þn J−ν ðzÞν = n π ∂ν

; n = 0, ± 1, ± 2, ± 3, . . .

(3:1:5)

Y− n ðzÞ = ð− 1Þn Yn ðzÞ Thus, in the case of the Bessel functions of the second kind, the mathematical operation – differentiation of the Bessel functions of the first kind with regard to the order emerges. The explicit expression for (3.1.5) is received by differentiation of the ascending series of Jν(z) from (2.9.1)

https://doi.org/10.1515/9783110681642-003

3.1 First Derivatives of the Bessel and Related Functions with Respect to the Order v

J± ν ðzÞ =

∞ X ð− 1Þk k =0

79

z 2k ± ν 2

k!Γðk ± ν + 1Þ

(3:1:6)

which gives ∞ z X z ν + 2k ψðν + k + 1Þ ∂ Jν ðzÞ ð− 1Þk = Jν ðzÞ ln − ∂ν 2 2 k!Γðν + k + 1Þ k =0

(3:1:7)

∞ z X z −ν + 2k ψð−ν + k + 1Þ ∂ J−ν ðzÞ ð− 1Þk = − J− ν ðzÞ ln + ∂ν 2 2 k!Γð−ν + k + 1Þ k =0

(3:1:8)

and

where the logarithmic derivative of the gamma function, the psi function is defined by  ∞  X d 1 d ΓðzÞ 1 1 ln ΓðzÞ = = −γ + − ψðzÞ = dz ΓðzÞ d z k+1 z+k k =0   1 = − γ − 2 ln 2 ψ 2 ψð1Þ = − γ = − 0.5772157 . . .   1 = − γ − 2 ln 2 ψ 2   3 = − γ − 2 ln 2 + 2 ψ 2

(3:1:9)

and γ denotes the Euler constant. In Figure 3.1, the first derivatives of the Bessel functions of the first kind with regard to the order are plotted. As can be observed, they change only for small values of the order, ν < 3. For larger values of ν, irrespective of the argument t, these derivatives tend to be zero. Three-dimensional plot of the first derivatives and tables of their values are presented in Part 2 of this book. The expressions in (3.1.7) and (3.1.8) serve to determine the first derivative of the Bessel function of the second kind with regard to the order ν ∂ Yν ðzÞ ∂ Jν ðzÞ ∂ J−ν ðzÞ = cotðπ νÞ − cosecðπ νÞ − π cosecðπ νÞY−ν ðzÞ ∂ν ∂ν ∂ν

(3:1:10)

The modified Bessel functions of the second kind Iν(z) and Kν(z) are linearly independent solutions of the following differential equation z2

d2 wðzÞ dwðzÞ +z ± ðz2 + ν2 ÞwðzÞ = 0 d z2 dz

(3:1:11)

80

3 Differentiation and Integration with Respect to the Order of the Bessel

1.0

7 7

6

8

0.0

𝜕J󰜈t 𝜕󰜈

3 –1.0

5

4

2 1

–2.0 0.0

1.0

2.0

󰜈

3.0

4.0

Figure 3.1: The first derivatives of the Bessel function of the first kind with regard to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 1.0; 5 – t = 2.0; 6 – t = 3.0; 7 – t = 4.0; 8 – t = 5.0

and similarly we have wðzÞ = A1 Iν ðzÞ + A2 I−ν ðzÞ ;   π I −ν ðzÞ − Iν ðzÞ Kν ðzÞ = 2 sinðπ νÞ

ν ≠ 0, ± 1, ± 2, ± 3, . . . (3:1:12)

and therefore for ν being an integer n " # ∂ ðI −ν ðzÞ − Iν ðzÞÞ π ∂ν ; Kn ðzÞ = ∂ 2 ∂ν ðsinðπ νÞÞ

n = 0, 1, 2, 3, . . .

ν =n

(3:1:13)

I − n ðzÞ = In ðzÞ Using the series expansion from (2.9.5) Iν ðzÞ =

∞ X

z 2 k + ν 2

k!Γðk + ν + 1Þ k =0

(3:1:14)

direct differentiation gives ∞  ν + 2 k z X ∂ Iν ðzÞ z ψðν + k + 1Þ = Iν ðzÞ ln − ∂ν 2 2 k!Γðν + k + 1Þ k =0

(3:1:15)

3.1 First Derivatives of the Bessel and Related Functions with Respect to the Order v

81

and   ∂ Kν ðzÞ π ∂ I −ν ðzÞ ∂ Iν ðzÞ = − π cotðπ νÞKν ðzÞ + cosecðπ νÞ − ∂ν 2 ∂ν ∂ν

(3:1:16)

As can be observed, in expressions (3.1.7), (3.1.8) and (3.1.15), the term-by-term differentiation is performed in two places, the ν – powers of variable z, and the orders ν included in the gamma functions. Introducing the series expansion of the Struve functions from (2.9.7) z 2k + ν + 1 2  ð− 1Þ  H ν ðzÞ = 3 3 Γ k + 2 Γ k+ν+ 2 k =0 ∞ X

k

(3:1:17)

it follows that z ν + 2k + 1  ∞ z X ψ ν + k + 32 ∂ H ν ðzÞ k 2  ð− 1Þ  = H ν ðzÞ ln − ∂ν 2 Γ k + 32 Γ ν + k + 32 k =0

(3:1:18)

and similarly from (2.9.10) Lν ðzÞ =

z 2k + ν + 1 2   3 3 Γ k + 2 Γ k+ν+ 2 k =0 ∞ X

(3:1:19)

we have   ∞ z ν + 2k + 1 z X ψ ν + k + 32 ∂ Lν ðzÞ 2  = Lν ðzÞ ln − ∂ν 2 Γ k + 32 Γ ν + k + 32 k =0

(3:1:20)

Considering that (2.9.19) berν ðzÞ =

     k ∞ z ν X cos π 3 ν + 4 k z2 4

2

k =0

k!Γðν + k + 1Þ

2

(3:1:21)

and (2.9.21) beiν ðzÞ =

   k ∞ z ν X sin π 3 ν + 4 k z2 4

2

k =0

k!Γðν + k + 1Þ

2

the derivatives of the Kelvin functions are z ∂ berν ðzÞ 3π = ln berν ðzÞ − beiν ðzÞ ∂ν 2 4      ∞ z ν X cos π 3 ν +4 4 k ψðν + k + 1Þ z2 k − 2 k!Γðν + k + 1Þ 2 k =0

(3:1:22)

(3:1:23)

82

3 Differentiation and Integration with Respect to the Order of the Bessel

and z ∂ beiν ðzÞ 3π = ln beiν ðzÞ + berν ðzÞ ∂ν 2 4      ∞ z ν X sin π 3 ν +4 4 k ψðν + k + 1Þ z2 k − 2 k!Γðν + k + 1Þ 2 k =0

(3:1:24)

In the case of other related Bessel functions, the series expansions are less suitable to determine desired derivatives with regard to the order because differentiation should be performed at more than three places in the corresponding series, and therefore other approaches are more preferable, as will be discussed later. The Bessel functions can also be expressed in terms of series of the Bessel functions having integer order z k + ν ∞ 1 X 2 Jk ðzÞ Jν ðzÞ = ΓðνÞ k = 0 k!ðk + νÞ

 k + ν ∞ ð− 1Þk z2 1 X Iν ðzÞ = Ik ðzÞ ΓðνÞ k = 0 k!ðk + νÞ

(3:1:25)

(3:1:26)

and therefore using ψðν + 1Þ = ψðνÞ +

1 ν

(3:1:27)

we have i ∂ Jν ðzÞ h z = ln − ψðν + 1Þ Jν ðzÞ ∂ν 2  k + ν ∞ X k z2 1 + Jk ðzÞ Γðν + 1Þ k = 1 k!ðk + νÞ2

(3:1:28)

i ∂ Iν ðzÞ h z = ln − ψðν + 1Þ Iν ðzÞ ∂ν 2  k + ν ∞ X ð− 1Þk k z2 1 + Ik ðzÞ Γðν + 1Þ k = 1 k!ðk + νÞ2

(3:1:29)

and

Brychkov [45] presented the first derivatives of the Bessel and related functions with respect to the order in a more closed form by using the generalized hypergeometric functions:

3.1 First Derivatives of the Bessel and Related Functions with Respect to the Order v

83

 2 ν π z2 ∂ Jν ðzÞ ½ Yν ðzÞ − cotðπ νÞJν ðzÞ f ðνÞ = ∂ν 2 ½Γðν + 1Þ2   z 1 z2 gðνÞ + Jν ðzÞ − ψðν + 1Þ + ln − 4ðν2 − 1Þ 2ν 2   1 f ðνÞ = 2 F3 ν, ν + ; ν + 1, ν + 1, 2 ν + 1; − z2 2   3 2 gðνÞ = 3 F4 1, 1, ; 2, 2, 2 − ν, ν + 2; − z 2

(3:1:30)

z 2 ν h i ∂ Iν ðzÞ π 2 K ðzÞ + ðzÞ f ðνÞ = cscðπ νÞI ν ν ∂ν 2 ½Γðν + 1Þ2   z 1 z2 gðνÞ + Iν ðzÞ − ψðν + 1Þ + ln − 4ðν2 − 1Þ 2ν 2   1 f ðνÞ = 2 F3 ν, ν + ; ν + 1, ν + 1, 2 ν + 1; z2 2   3 gðνÞ = 3 F4 1, 1, ; 2, 2, 2 − ν, ν + 2; z2 2

(3:1:31)

and

Brychkov also derived more lengthy expressions for derivatives of Yν(z) and Kν(z) by using the 3F4 hypergeometric functions, and for derivatives of berν(z) and beiν(z) the 4F7 hypergeometric functions. Using these expressions, it is possible to obtain derivatives of kerν(z) and keiν(z) functions by applying interrelations between the Kelvin functions as given in (2.1.26) and (2.1.27). Recently, González-Santander [53, 54, 86, 87] investigated derivatives of the Bessel and the Kelvin functions and expressed them in terms of the hypergeometric and the Meijer-G functions. Differentiation of the recurrence formulas between the Bessel functions Jν(z) and Yν(z) gives interrelation between different orders, for example from (2.2.13) Cν + 1 ðzÞ =

2ν Cν ðzÞ − Cν −1 ðzÞ z

(3:1:32)

we have ∂ Cν + 1 ðzÞ 2ν ∂Cν ðzÞ ∂ Cν − 1 ðzÞ 2 = − + Cν ðzÞ ∂ν z ∂ν ∂ν z

(3:1:33)

Similarly, from (2.3.6) and (2.3.21) Iν + 1 ðzÞ = Iν − 1 ðzÞ −

2ν Iν ðzÞ z

(3:1:34)

84

3 Differentiation and Integration with Respect to the Order of the Bessel

Kν + 1 ðzÞ = Kν − 1 ðzÞ +

2ν Kν ðzÞ z

(3:1:35)

it follows that ∂ Iν + 1 ðzÞ 2ν ∂ Iν ðzÞ ∂ Iν − 1 ðzÞ 2 =− + − Iν ðzÞ ∂ν z ∂ν ∂ν z

(3:1:36)

∂ Kν + 1 ðzÞ 2ν ∂ Kν ðzÞ ∂ Kν − 1 ðzÞ 2 = − + Kν ðzÞ ∂ν z ∂ν ∂ν z

(3:1:37)

and

In the case of the Struve functions the recurrence formulas are (2.4.4) and (2.4.22) z ν 2ν H ν + 1 ðzÞ = (3:1:38) H ν ðzÞ − H ν − 1 ðzÞ + pffiffiffi 2 z π Γ ν + 32 z ν 2ν (3:1:39) Lν + 1 ðzÞ = Lν − 1 ðzÞ − Lν ðzÞ − pffiffiffi 2 z π Γ ν + 32 and therefore ∂ H ν + 1 ðzÞ 2ν ∂ H ν ðzÞ ∂ H ν − 1 ðzÞ 2 = − + H ν ðzÞ ∂ν z ∂ν ∂ν z z ν     z 3 pffiffiffi 2 + ln −ψ ν+ 2 2 π Γ ν + 32

(3:1:40)

∂ Lν + 1 ðzÞ ∂ Lν − 1 ðzÞ 2ν ∂ Lν ðzÞ 2 = − − Lν ðzÞ ∂ν ∂ν z ∂ν z z ν      3 z pffiffiffi 2 − ln + ψ ν+ 2 2 π Γ ν + 32

(3:1:41)

and

The Lommel differential equation z2

d2 wðzÞ dwðzÞ +z + ðz2 − ν2 ÞwðzÞ = zμ d z2 dz

(3:1:42)

includes two parameters (or variables) ν and μ, but here only ν is of importance because the homogenous equation is satisfied by the Bessel functions of the order ν and therefore differentiation with regard to it will only be performed. If two parameters ν and μ are identical, ν = μ, then the Lommel functions can be expressed in terms of the Neumann and Struve functions

3.1 First Derivatives of the Bessel and Related Functions with Respect to the Order v

  pffiffiffi 1 H ν ðzÞ sν, ν ðzÞ = 2 πΓ ν+ 2   pffiffiffi 1 ν−1 ½H ν ðzÞ − Yν ðzÞ Sν, ν ðzÞ = 2 πΓ ν+ 2 ν−1

85

(3:1:43) (3:1:44)

which gives       pffiffiffi ∂ sν, ν ðzÞ 1 ∂ H ν ðzÞ 1 H ν ðzÞ = 2ν − 1 π Γ ν + + ln 2 + ψ ν + ∂ν 2 ∂ν 2

(3:1:45)

and ∂ Sν, ν ðzÞ = ∂ν

  pffiffiffi ∂ H ν ðzÞ ∂ Yν ðzÞ 1 − + 2ν − 1 π Γ ν + ∂ν ∂ν 2     1 ½H ν ðzÞ − Yν ðzÞ ln 2 + ψ ν + 2

(3:1:46)

In general case, from (2.9.11) sμ, ν ðzÞ =

 2k   ∞ ð− 1Þk z2 Γ μ −ν2 + 1 Γ μ + 2ν + 1 zμ+1 X   4 k = 0 Γ μ −ν + 2k + 3 Γ μ + ν + 2k + 3 2

(3:1:47)

2

differentiation yields  2k   ∞ ∂ sμ, ν ðzÞ z μ + 1 X ð− 1Þk z2 Γ μ −ν2 + 1 Γ μ + 2ν + 1   = ak ∂ν 4 k = 0 Γ μ −ν + 2k + 3 Γ μ + ν + 2k + 3 2

2

h    i  ak = ψ μ −ν2 + 1 + ψ μ + 2ν + 1 − ψ μ −ν +22k + 3 − ψ μ −ν +22k + 3

(3:1:48)

Since Sλ, ν ðzÞ = sμ, ν ðzÞ + Aμ, ν Bμ, ν     μ −ν + 1 μ+ν+1 Γ Aμ, ν = 2μ − 1 Γ 2 2       π ðμ −νÞ π ðμ −νÞ Jν ðzÞ − cos Yν ðzÞ Bμ, ν = sin 2 2

(3:1:49)

86

3 Differentiation and Integration with Respect to the Order of the Bessel

we have      ∂ Sλ, ν ðzÞ ∂sμ, ν ðzÞ 1 μ+ν+1 μ −ν + 1 −ψ = + Aμ, ν Bμ, ν ψ ∂ν 2 2 2 ∂ν    π ðμ −νÞ ∂ Jν ðzÞ π + Aμ, ν sin − Yν ðzÞ 2 ∂ν 2    π ðμ −νÞ ∂ Yν ðzÞ π − Aμ, ν cos + Jν ðzÞ 2 ∂ν 2

(3:1:50)

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order ν: Particular Values of the Order ν Since the Bessel functions of integer order can be expressed in the form of finite series, their derivatives with regard to the order are the finite sums of the corresponding Bessel and Hankel functions [18]     n − 1  k − n ∂ Jν ðzÞ π n! X z Jk ðzÞ (3:2:1) = ð± 1Þn Yn ðzÞ ± ∂ν ν = ± n 2 2 k =0 2 k!ðn − kÞ     n − 1  k − n ∂ Yν ðzÞ π n! X z Yk ðzÞ (3:2:2) = ð± 1Þn − Jn ðzÞ ± ∂ν ν = ± n 2 2 k =0 2 k!ðn − kÞ " # " # ðjÞ ðjÞ n − 1  k − n Hk ðzÞ ∂ Hν ðzÞ n! X z j π i ðjÞ = ð− 1Þ H ðzÞ + k!ðn − kÞ ∂ν 2 n 2 k =0 2 (3:2:3) ν= n

j = 1, 2 " # ðjÞ ∂ Hν ðzÞ ∂ν

= ð− 1Þj

+n

ν= −n

"

π i ðjÞ H ðzÞ 2 n

n − 1  k − n Hk ðzÞ n! X z − ð− 1Þ k!ðn − kÞ 2 k =0 2 ðjÞ

n

#

(3:2:4) ; j = 1, 2

and " #   n − 1  k − n ∂ Iν ðzÞ n! X z ð− 1Þk Ik ðzÞ n = ð− 1Þ − Kn ðzÞ ± ∂ν ν = ± n 2 k =0 2 k!ðn − kÞ     n − 1  k − n ∂ Kν ðzÞ n! X z Kk ðzÞ = ± ∂ν ν = ± n 2 k =0 2 k!ðn − kÞ

(3:2:5)

(3:2:6)

Derivatives with respect to the integer order of the Lommel, Anger and Struve functions are

87

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v

 n − 1 1 − n 1 − λ − n   n!Γ 1 − λ2 − n X z ∂ Sλ, ν ðzÞ 2 = 1 − λ + n Sλ + n − k, k ðzÞ ðn − kÞ k! ∂ν 2Γ 2 ν =n k=0      1 1−λ+n 1−λ−n −ψ Sλ, ν ðzÞ − ψ 2 2 2

(3:2:7)

  n − 1  n − k ∂ J ν ðzÞ π n! X 2 Jk ðzÞ = H n ðzÞ + ∂ν ν = n 2 2 k =0 z k!ðn − kÞ   n−1 n−1 Γ k + 21 z n − 2k − 1 ð− 1Þn X ð− 1Þk Γ n + 2k + 1 z k 1X   + − z k = 0 Γ n − 2k + 1 2 k = 0 Γ n − k + 21 2 2 (3:2:8) and   n − 1  n − k ∂ J ν ðzÞ π ð− 1Þn − 1 n! X 2 Jk ðzÞ = H − n ðzÞ + ∂ν ν = −n 2 2 z k!ðn − kÞ k =0  n−1 Γ n + 2k + 1 z k ð− 1Þn X  + z k = 0 Γ n − 2k + 1 2

(3:2:9)

#  2 n − k "     n−1 k−1 Γ j + 21 ∂ Eν ðzÞ π n! X 1 X z k − 2j − 1 z  = Jn ðzÞ + − H k ðzÞ + ∂ν ν = n 2 2 k = 0 k!ðn − kÞ π j = 0 Γ k − j + 21 2 n + k + 1  k + 1 k − 1 n−1 X 1 X 2 1 k n Γ + ½ð− 1Þ + ð− 1Þ  n − 2k + 1 − 2π k =0 z n − k + 2j + 1 Γ 2 j =0 (3:2:10)  2 n − k   n−1 −z ∂ Eν ðzÞ π n! X = ð− 1Þn Jn ðzÞ + H − k ðzÞ ∂ν ν = − n 2 2 k = 0 k!ðn − kÞ n + k + 1  k + 1 k − 1 n−1 X 1 X 2 1 k n Γ ½ð− 1Þ + ð− 1Þ  n − 2k + 1 2π k = 0 z n − k + 2j + 1 Γ 2 j =0

(3:2:11)

The analogous expressions for derivatives with respect to the integer order of the Kelvin functions are   ∂ berν ðzÞ π = − bein ðzÞ − kern ðzÞ ∂ν 2 ν =n z k − n   n−1 n! X 5 π ðk − nÞ 2 berk ðzÞ cos + (3:2:12) 2 k = 0 k!ðn − kÞ 4 z k − n   n−1 n! X 5 π ðk − nÞ 2 + beik ðzÞ sin 2 k = 0 k!ðn − kÞ 4

88

3 Differentiation and Integration with Respect to the Order of the Bessel

  ∂ beiν ðzÞ π = bern ðzÞ − kein ðzÞ ∂ν 2 ν =n z k − n   n−1 n! X 5 π ðk − nÞ 2 + beik ðzÞ cos 2 k = 0 k!ðn − kÞ 4 z k − n   n−1 n! X 5 π ðk − nÞ 2 berk ðzÞ sin − 2 k = 0 k!ðn − kÞ 4

(3:2:13)

and z k − n     n−1 ∂ kerν ðzÞ π n! X 3π ðk − nÞ 2 kerk ðzÞ = kein ðzÞ + cos ∂ν 2 2 k = 0 k!ðn − kÞ 4 ν =n z k − n   n−1 n! X 3 π ðk − nÞ 2 keik ðzÞ sin − 2 k = 0 k!ðn − kÞ 4

z k − n     n−1 ∂ keiν ðzÞ π n! X 3 π ðk − nÞ 2 kerk ðzÞ = − kern ðzÞ + sin ∂ν 2 2 k = 0 k!ðn − kÞ 4 ν =n z k − n   n−1 n! X 3 π ðk − nÞ 2 + keik ðzÞ cos 2 k = 0 k!ðn − kÞ 4

(3:2:14)

(3:2:15)

Derivatives of the Struve functions can be expressed for v = ± n only by using the Meijer-G functions !      1=2 1=2 ∂ H ν ðzÞ π 1 2 n 32 z2  = − Jn ðzÞ + G24 4  1=2 , 1=2 , n , 0 ∂ν 2 2π z ν=n   n − 2 k − 1     n−1 Γ k + 21 z2 1 X z 1 (3:2:16)  + ln − ψ n − k + π k =0 2 2 Γ n − k + 21  k − n n−1 ð− 1Þk z2 n! X + H − k ðzÞ ; Re z ≥ 0 2 k = 0 k!ðn − kÞ !  1=2    n 2 1=2 ∂ H ν ðzÞ 2 n+1 π n 1 32 z  = ð− 1Þ G24 Jn ðzÞ + ð− 1Þ 4  1=2 , 1=2 , n , 0 ∂ν 2 2π z ν =−n  z k − n n−1 −2 n! X − H − k ðzÞ ; Re z ≥ 0 2 k = 0 k!ðn − kÞ (3:2:17)

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v

89

and !      1=2 ∂ Lν ðzÞ ð− 1Þn − 1 2 n 42 z2  1=2 n = ð− 1Þ Kn ðzÞ + G24 2π2 4  0 , 1=2 , 1=2 , n ∂ν ν = n z   n − 2k − 1     n−1 ð− 1Þk Γ k + 21 z2 1 X z 1  ln − ψ n − k + − π k =0 2 2 Γ n − k + 21  k−n n−1 − z2 n! X + L − k ðzÞ ; Re z ≥ 0 2 k = 0 k!ðn − kÞ !      1=2 ∂ Lν ðzÞ ð− 1Þn − 1 2 n 42 z2  1=2 n = ð− 1Þ Kn ðzÞ + G24 2 π2 4  0, 1=2 1=2, n ∂ν ν = − n z  z k − n n−1 −2 n! X L − k ðzÞ − 2 k = 0 k!ðn − kÞ

(3:2:18)

(3:2:19)

Re z ≥ 0 Derivatives with respect to the order for positive and negative integer values of ν can be determined using general formulas given above, however the preferable technique is to obtain them consecutively, by applying the corresponding recurrence formulas that exist between the Bessel or related functions. The starting point is evident at ν = 0 and ν = ± 1, and therefore a significant effort was directed to obtain them. Functional expressions for derivatives with respect to the order, especially for its zero value are well known and they are presented in the mathematical literature [9, 18]. There are a number of techniques to derive such expressions as is illustrated below. The consecutive derivation of derivatives in the simplest case of the Bessel functions of the first and second kind follows from (2.2.13):   ∂ Jν ðzÞ π = Y0 ðzÞ (3:2:20) ∂ν ν = 0 2     ∂ J ν ðzÞ ∂ J ν ðzÞ 2 + = J0 ðzÞ (3:2:21) ∂ν ν = −1 ∂ν ν =1 z and   ∂ J ν ðzÞ − ∂ν ν = 1

  ∂ J ν ðzÞ = π Y1 ðzÞ ∂ν ν = −1

Combining these equations gives   ∂ J ν ðzÞ 1 π = J0 ðzÞ − Y1 ðzÞ ∂ν ν = −1 z 2

(3:2:22)

(3:2:23)

90

3 Differentiation and Integration with Respect to the Order of the Bessel

  ∂ J ν ðzÞ 1 π = J0 ðzÞ + Y1 ðzÞ ∂ν ν =1 z 2 Similarly, we have   ∂ Y ν ðzÞ 1 π = Y0 ðzÞ + J1 ðzÞ ∂ν ν = −1 z 2   ∂ Yν ðzÞ π = − J0 ðzÞ ∂ν ν = 0 2   ∂ Y ν ðzÞ 1 π = Y0 ðzÞ − J1 ðzÞ ∂ν ν =1 z 2

(3:2:24)

(3:2:25)

(3:2:26)

(3:2:27)

Since the Hankel functions are expressed simply by Hνð1Þ ðzÞ = Jν ðzÞ + i Yν ðzÞ

(3:2:28)

Hνð2Þ ðzÞ = Jν ðzÞ − i Yν ðzÞ

(3:2:29)

Their first derivatives with respect to the order are " # ð1Þ ð1Þ ∂H ν ðzÞ H ðzÞ π i ð1Þ = 0 + H ðzÞ ∂ν z 2 1 "

"

and "

"

"

ð1Þ

∂H ν ðzÞ ∂ν ð1Þ

∂H ν ðzÞ ∂ν

#

=− #

ð2Þ

∂H ν ðzÞ ∂ν ð2Þ

∂H ν ðzÞ ∂ν ð2Þ

∂H ν ðzÞ ∂ν

(3:2:30)

ν = −1

ν =0

π i ð1Þ H ðzÞ 2 0

(3:2:31)

ð1Þ

= ν=1

H0 ðzÞ π i ð1Þ − H ðzÞ z 2 1

#

ð2Þ

= #

#

(3:2:32)

ν = −1

H0 ðzÞ π i ð2Þ − H ðzÞ z 2 1

=

π i ð2Þ H ðzÞ 2 0

=

H0 ðzÞ π i ð2Þ H ðzÞ + 2 1 z

ν =0

(3:2:33)

(3:2:34)

ð2Þ

ν=1

(3:2:35)

In the case of the modified Bessel function of the first kind Iν(z) we have from (2.3.6)

91

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v

    ∂ I ν ðzÞ ∂ I ν ðzÞ 2 − = I0 ðzÞ ∂ν ν = −1 ∂ν ν = 1 z I −ν ðzÞ = Iν ðzÞ +

2 sinðπ νÞ Kν ðzÞ π

(3:2:36) (3:2:37)

and differentiation of (3.2.27) gives −

∂ I −ν ðzÞ ∂ I ν ðzÞ 2 ∂K ν ðzÞ = + 2 cosðπ νÞ Kν ðzÞ + sinðπ νÞ ∂ν ∂ν π ∂ν

By introducing ν = 1, it follows that     ∂ I ν ðzÞ ∂ I ν ðzÞ + = 2K1 ðzÞ ∂ ν ν = −1 ∂ν ν =1 and finally   ∂ I ν ðzÞ 1 = I0 ðzÞ + K1 ðzÞ ∂ν ν = −1 z   ∂ Iν ðzÞ = − K0 ðzÞ ∂ν ν = 0   ∂ I ν ðzÞ 1 = K1 ðzÞ − I0 ðzÞ ∂ν ν =1 z

(3:2:38)

(3:2:39)

(3:2:40)

(3:2:41)

(3:2:42)

Considering that, for the modified Bessel function of the second kind we have Kν ðzÞ = K −ν ðzÞ Kν − 1 ðzÞ − Kν + 1 ðzÞ = −

(3:2:43) 2ν Kν ðzÞ z

(3:2:44)

and therefore from (3.2.43) and (3.2.44) it follows that ∂ Kν ðzÞ ∂ Kν − 1 ðzÞ =− ∂ν ∂ν

(3:2:45)

∂ K ν − 1 ðzÞ ∂ K ν + 1 ðzÞ 2 2 ν ∂ K ν ðzÞ + = Kν ðzÞ + ∂ν ∂ν z z ∂ν

(3:2:46)

Introducing ν = 0 in these equations gives     ∂ K ν ðzÞ 1 ∂ K ν ðzÞ = K0 ðzÞ = − ∂ν ν = 1 z ∂ν ν = −1   ∂ K ν ðzÞ =0 ∂ν ν = 0

(3:2:47)

(3:2:48)

92

3 Differentiation and Integration with Respect to the Order of the Bessel

Similarly, by using series expansions given in Section 3.1 and by differentiating the recurrence expressions of other related Bessel functions, it is possible to obtain their derivatives with respect to the order, initially for ν = 0 and ν = ± 1 and after this for negative and positive integers. Final results for these values of ν, in the case of the integral Bessel functions Jiν(z), Yiν(z) and Kiν(z) are known from the Brychkov and Geddes paper [44]   ∂ Ji ν ðzÞ 1 π = − J0 ðzÞ + Yi1 ðzÞ − Ji1 ðzÞ (3:2:49) ∂ν ν = −1 z 2   ∂ Ji ν ðzÞ π = − Yi0 ðzÞ (3:2:50) ∂ν ν = 0 2   ∂ Ji ν ðzÞ 1 π = − J0 ðzÞ − Yi1 ðzÞ − Ji1 ðzÞ (3:2:51) ∂ν ν = 1 z 2 and   ∂ Yi ν ðzÞ 1 π = − Y0 ðzÞ − Ji1 ðzÞ − Yi1 ðzÞ ∂ν z 2 ν = −1   ∂ Yi ν ðzÞ π = Ji0 ðzÞ ∂ν 2 ν=0   ∂ Yi ν ðzÞ 1 π = − Y0 ðzÞ + Ji1 ðzÞ − Yi1 ðzÞ ∂ν z 2 ν=1 and finally for the integral modified Bessel function of the second kind   ∂ Ki ν ðzÞ 1 π = K0 ðzÞ − Ki1 ðzÞ ∂ν z 2 ν = −1   ∂ Ki ν ðzÞ =0 ∂ν ν=0   ∂ Ki ν ðzÞ 1 π = − K0 ðzÞ + Ki1 ðzÞ ∂ν z 2 ν=1

(3:2:52)

(3:2:53)

(3:2:54)

(3:2:55)

(3:2:56)

(3:2:57)

A number of additional expressions for derivatives with respect to the order of the Kelvin, Struve, Anger, Weber and Lommel functions at ν = 0 or for other values of the order are also known and they are presented here   ∂ ber ν ðzÞ π = − beiðzÞ − kerðzÞ (3:2:58) ∂ν 2 ν=0   ∂ bei ν ðzÞ π = berðzÞ − keiðzÞ ∂ν 2 ν=0

(3:2:59)

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v



∂ H ν ðzÞ ∂ν

 2 1 π 1 32 z  = − J0 ðzÞ + 2 G24  4  1=2, 1, 2 πz ν=0



1

∂ Lν ðzÞ ∂ν

 2 1 z  1 = K0 ðzÞ − 2 G42  24 4  1=2, 1, πz ν=0



! (3:2:60)

1, 1=2

Re z ≥ 0



93

1

! (3:2:61)

1, 1=2

Re z ≥ 0   ∂ J ν ðzÞ π = H 0 ðzÞ ∂ν ν = 0 2   ∂ E ν ðzÞ π = J0 ðzÞ ∂ν ν = 0 2   ∂ S0, ν ðzÞ = 2 S − 2, 1 ðzÞ ∂ν ν = 1

(3:2:62)

(3:2:63)

(3:2:64)

Formulas for half odd integer order, v = ± n ± 1/2, n = 0, 1, 2, 3, . . . are much more longer than derivatives with the integer orders, and they can be found in the Brychkov publications [11, 44, 45]. In the mathematical literature, special attention was paid to the case v = ± 1/2. Derivatives with respect to the order for half odd orders were derived by various techniques and they can be expressed in terms of trigonometric and exponential integrals. The applied methods will be discussed later, here the final results are only listed for the Bessel functions of the first and the second kind rffiffiffiffiffiffi   ∂ Jν ðzÞ 2 ½cos z Cið2zÞ + sin z Sið2 zÞ (3:2:65) = ∂ν ν = − 1=2 πz   ∂ Jν ðzÞ ∂ν ν

=

rffiffiffiffiffiffi 2 ½sin z Cið2zÞ − cos z Sið2 zÞ = πz 1=2

(3:2:66)

and rffiffiffiffiffiffi   ∂ Yν ðzÞ 2 =− fsin z Cið2zÞ + cos z½π − Sið2zÞg ∂ν ν = − 1=2 πz

(3:2:67)

rffiffiffiffiffi   ∂ Yν ðzÞ 2 fcos z Cið2 zÞ − sin z½π − Sið2 zÞg = ∂ν ν = 1=2 πz

(3:2:68)

For the modified Bessel functions of the first and the second kind we have   ∂ Iν ðzÞ 1 = pffiffiffiffiffiffiffiffiffi ½ez Eið− 2 zÞ + e−z Eið2zÞ ∂ν ν = − 1=2 2π z

(3:2:69)

94

3 Differentiation and Integration with Respect to the Order of the Bessel

  ∂ Iν ðzÞ 1 = pffiffiffiffiffiffiffiffiffi ½ez Eið− 2 zÞ − e−z Eið2zÞ ∂ν ν = 1=2 2πz and rffiffiffiffiffi   ∂ Kν ðzÞ π z = e Eið− 2zÞ ∂ν ν = − 1=2 2z rffiffiffiffiffi   ∂ Kν ðzÞ π z =− e Eið− 2 zÞ ∂ν ν = 1=2 2z

(3:2:70)

(3:2:71)

(3:2:72)

Derivatives with respect to the order of the Struve functions of the first and the second kind are rffiffiffiffiffiffi(   ) ∂ H ν ðzÞ 2 (3:2:73) = cos z ½Sið2 zÞ − 2SiðzÞ − sin z ½Cið2 zÞ − 2CiðzÞ ∂ν πz ν = − 1=2 rffiffiffiffiffiffi   z ∂ H ν ðzÞ 2 γ + ln = + cos z ½Cið2 zÞ − 2CiðzÞ ∂ν πz 2 ν = 1=2  + sin z ½Sið2 zÞ − 2SiðzÞ   ∂ Lν ðzÞ 1 = pffiffiffiffiffiffiffiffiffi f e−z ½Eið2 zÞ − 2EiðzÞ − e z ½Eið− 2zÞ − 2Eið− zÞg ∂ν ν = − 1=2 2πz   z i ∂ Lν ðzÞ 1 n h + e−z ½Eið2zÞ − 2 EiðzÞ = − pffiffiffiffiffiffiffiffiffi 2 γ + ln 2 ∂ν ν = 1=2 2πz o + ez ½Eið− 2 zÞ − 2Eið− zÞ

(3:2:74)

(3:2:75)

(3:2:76)

where the sine, cosine and exponential integrals are defined by ðz SiðzÞ =

sin t dt t

(3:2:77)

0 ∞ ð

CiðzÞ = −

ðz cos t cos t − 1 dt = γ + ln z + dt t t

z

(3:2:78)

0

ðz EiðzÞ = γ + ln z +

et − 1 dt t

(3:2:79)

0

ðz − Eið−zÞ = − γ − ln z + 0

1 − e− t dt t

(3:2:80)

95

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v

As pointed out above, the derivation of derivatives with respect to the order of the Bessel functions in the special case, v = ± 1/2, was performed in a number of different ways. The results presented here were taken mainly from the Oberhettinger paper [56]. In his derivations, the integral representation of the modified Bessel function of the second kind was used as a starting point. Alternative procedures that are based on operational calculus, interrelations between Bessel functions and recurrence relations are illustrated below. The substitution formula of the Laplace transformation is [32] ∞ ð

FðsÞ = Lff ðtÞg =

L

−1

e− s t f ðtÞ dt

0

∞  ð  x + 1=2 pffiffiffi Jx + 1=2 ðtÞ 1 t 2 f ðxÞ dx + 1Þ = π F½lnðs 2 Γðx + 1Þ ðs + 1Þ 2

(3:2:81)

0

If the transform-inverse pair includes the derivative of the delta Dirac function δʹ(t), then (3.2.81) can be written as   L δ′ðtÞ = s ) ∞

 ð ( x + 1=2 pffiffiffi Jx + 1=2 ðtÞ 1 t −1 2 δ′ðxÞ dx lnðs + 1Þ = π L Γðx + 1Þ ðs2 + 1Þ 2 (3:2:82) 0 (  ) pffiffiffi d t x + 1=2 Jx + 1=2 ðtÞ =− π Γðx + 1Þ dx 2 x =0

The inverse on the left-hand side of (3.2.82) is known [41]

     lnðs2 + 1Þ t = cos t Sið2tÞ − sin t γ + ln + Cið2 tÞ L−1 ðs2 + 1Þ 2

(3:2:83)

and differentiation of the integrand in (3.2.82) gives (  )  x + 1=2      t ∂Jx + 1=2 ðtÞ d t x + 1=2 Jx + 1=2 ðtÞ t 2 − ψðx + 1Þ Jx + 1=2 ðtÞ + = ln Γðx + 1Þ ∂x Γðx + 1Þ dx 2 2 ψðx + 1Þ =

Γ′ðx + 1Þ Γðx + 1Þ (3:2:84)

96

3 Differentiation and Integration with Respect to the Order of the Bessel

Considering that rffiffiffiffiffi 2 sin t J1=2 ðtÞ = πt

(3:2:85)

ψð1Þ = − γ = − 0.57721566... and introducing x = 0 in (3.2.84) the expected result is reached. rffiffiffiffiffi

 ∂ Jν ðtÞ 2 = fsin t Cið2 tÞ − cos t Sið2 tÞg ∂ν ν = 1=2 πt

(3:2:86)

Following the Müller approach [73], it is possible to obtain the derivative with respect to the order for ν = − 1/2, by using the recurrence relation of the Bessel functions of the first kind dJν ðtÞ ν = Jν − 1 ðtÞ − Jν ðtÞ dt t Differentiation of (3.2.87) when ν is a variable yields   d ∂ Jν ðtÞ ∂ Jν − 1 ðtÞ ν ∂ Jν ðtÞ Jν ðtÞ = − − dt ∂ν ∂ν t ∂ν t and by introducing ν = 1/2 into (3.2.88) we have (  " #)     J1=2 ðtÞ ∂ Jν ðtÞ d ∂ Jν ðtÞ 1 ∂Jν ðtÞ = + + 2 ∂ν ν = − 1=2 dt ∂ν ν = 1=2 t ∂ν ν = 1=2 The first term in the parenthesis is (rffiffiffiffiffi" #) ∞   ð ð2t d ∂ Jν d 2 cos x sin x − sin t = dx − cos t dx dt ∂ν ν = 1=2 dt πt x x 2t 0 

   sin t cos Cið2tÞ + sin t + = cos t − Sið2tÞ t t

(3:2:87)

(3:2:88)

(3:2:89)

(3:2:90)

and two additional terms are known and therefore once again we have the expected result rffiffiffiffiffi   ∂ Jν ðtÞ 2 (3:2:91) = fcos t Cið2 tÞ + sin t Sið2 tÞg ∂ν ν = − 1=2 πt As has been demonstrated by Petiau [23], derivatives with respect to the order of the Bessel function of the second kind are available from the interrelations between Jν(t) and Yν(t) functions

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v

  cosðπ νÞJν ðtÞ − J −ν ðtÞ 1 ∂ Y −ν ðtÞ Jν ðtÞ = = − cosðπ νÞYν ðtÞ sinðπ νÞ sinðπ νÞ ∂ν ∂ Yν ðtÞ ∂ Jν ðtÞ ∂ J −ν ðtÞ = cosðπ νÞ − cscðπ νÞ − π cscðπ νÞ Y−ν ðtÞ ∂ν ∂ν ∂ν Introducing ν = 1/2 into (3.2.92) gives the already known result     ∂ Yν ðtÞ ∂ Jν ðtÞ = − π Y−1=2 ðtÞ ∂ν ∂ν ν = − 1=2 ν = 1=2 rffiffiffiffiffi 2 Y − 1=2 ðtÞ = sin t πt rffiffiffiffiffi   ∂ Yν ðtÞ 2 = fcos t Cið2 tÞ + sin t ½Sið2 tÞ − πg ∂ν πt ν = 1=2 Similarly, if ν = − 1/2 is introduced into (3.2.92), we have     ∂ Yν ðtÞ ∂ Jν ðtÞ =− + π Y1=2 ðtÞ ∂ν ∂ν ν = − 1=2 ν = − 1=2 rffiffiffiffiffi 2 Y1=2 ðtÞ = cos t πt rffiffiffiffiffi   ∂ Yν ðtÞ 2 fcos t ½Sið2 tÞ − π − sin t Cið2tÞg = ∂ν π t ν = 1=2

97

(3:2:92)

(3:2:93)

(3:2:94)

The interrelation between the Bessel function Jν(t) and the modified Bessel function of the first kind Iν(t) Iν ðtÞ = e−i π ν=2 Jν ði tÞ

(3:2:95)

permits to obtain derivatives with respect to the order from (3.2.95) ∂ Iν ðtÞ πi ∂ Jν ði tÞ = − Iν ðtÞ + e − i π ν=2 ∂ν 2 ∂ν

(3:2:96)

Taking into account that for ν = 1/2 we have sinðitÞ = i sinh t

;

cosði tÞ = cosh t

i SiðitÞ = ½Eið2tÞ + E1 ð2tÞ 2 1 1 CiðitÞ = ½Eið2tÞ − E1 ð2tÞ = ½Eið2tÞ + Eið− 2tÞ 2 2 rffiffiffiffiffi rffiffiffiffiffi 2 2 sinh t ; I − 1=2 ðtÞ = cosh t I1=2 ðtÞ = πt πt

(3:2:97)

98

3 Differentiation and Integration with Respect to the Order of the Bessel

and e

− i π=4

rffiffiffiffiffi   ∂ Jν ði tÞ 2 fsinði tÞCið2itÞ − cosði tÞ Sið2i tÞg = −i ∂ν πt ν = 1=2 rffiffiffiffiffi  2 π i sinh t − et E1 ð2 tÞ − e− t Eið2 tÞ = πt

the derivative with respect to the order for ν = 1/2 is    ∂ Iν ðtÞ 1  = pffiffiffiffiffiffiffiffi et Eið− 2 tÞ − e− t Eið2tÞ ∂ν ν = 1=2 2π t Similarly, it is possible to obtain for ν = −1/2    ∂ Iν ðtÞ 1  = pffiffiffiffiffiffiffiffi et Eið− 2tÞ + e− t Eið2 tÞ ∂ν ν = − 1=2 2πt

(3:2:98)

(3:2:99)

(3:2:100)

The modified Bessel functions of the first and second kind, Iν(t) and Kν(t), are also interrelated Kν ðtÞ =

π ½I −ν ðtÞ − Iν ðtÞ 2 sinðπ νÞ

and differentiation of (3.2.101) with respect to the order ν gives

    ∂ Kν ðtÞ π ∂I −ν ðtÞ ∂Iν ðtÞ + = − π cotðπ νÞKν ðtÞ − cosecðπ νÞ ∂ν 2 ∂ν ∂ν Introducing derivatives for ν = ± 1/2, from (3.2.99) and (3.2.100) we have rffiffiffiffiffi   ∂ Kν ðtÞ π t =∓ e Eið− 2tÞ ∂ν 2t ν = ± 1=2

(3:2:101)

(3:2:102)

(3:2:103)

Struve function of the first kind Hν(t) is interrelated with the Bessel function of the second kind Yν(t) in the following way ∞  ν ð 2 z ν − 1=2 e−z t ð1 + t2 Þ dt H ν ðzÞ = Yν ðzÞ + pffiffiffi 2 1 π Γðν + 2Þ (3:2:104) 0 1 Re ν > − 2 Differentiation of (3.2.105) with respect to the order ν gives

99

3.2 First Derivatives of the Bessel and Related Functions with Respect to the Order v

   ∂ H ν ðzÞ ∂ Yν ðzÞ z 1 = + ln − ψðν + Þ ½H ν ðzÞ − Yν ðzÞ ∂ν ∂ν 2 2 ∞  z ν ð 2 ν − 1=2 + pffiffiffi 2 1 e−z t ð1 + t2 Þ lnð1 + t2 Þ dt π Γðν + 2Þ

(3:2:105)

0

For ν = 1/2 from (3.2.105) it follows that     h z i ∂ H ν ðzÞ ∂ Yν ðzÞ = + ln − ψð1Þ ½H 1=2 ðzÞ − Y1=2 ðzÞ ∂ν ∂ν 2 ν = 1=2 ν = 1=2 rffiffiffiffiffi ∞ ð 2z + e−z t lnð1 + t2 Þ dt π rffiffiffiffiffiffi 2 H 1=2 ðzÞ = ð1 − cos zÞ πz

(3:2:106)

0

The Laplace transform of the integral in (3.2.107) is known [36] ∞ ð

e−z t lnð1 + t2 Þ dt = −

h 2n π io cos z CiðzÞ + sin z SiðzÞ − z 2

(3:2:107)

0

and therefore the derivative with respect to the order ν is rffiffiffiffiffiffi   ∂ Hν 2 I = ∂ν ν = 1=2 πz n z o I = γ + ln + cos z½Cið2zÞ − 2CiðzÞ + sin z ½Sið2zÞ − 2SiðzÞ 2

(3:2:108)

If recurrence relations of the Struve functions are differentiated with regard to the argument z and the order ν       d ∂ H ν ðzÞ ∂ H ν − 1 ðzÞ ν ∂ H ν ðzÞ 1 = − − H ν ðzÞ (3:2:109) dz ∂ν ∂ν z ∂ν z and taking into account that       ∂ H ν − 1 ðzÞ ∂ H ν + 1 ðzÞ 2 ν ∂ H ν ðzÞ 2 + = + H ν ðzÞ ∂ν ∂ν z ∂ν z z ν     z 3 + pffiffiffi 2 ln − ψ ν + 2 2 π Γ ν + 32 it is possible to evaluate the derivative with respect to the order for ν = − 1/2

(3:2:110)

100

3 Differentiation and Integration with Respect to the Order of the Bessel

rffiffiffiffiffiffi   ∂ Hν 2 I = ∂ν ν = − 1=2 πz

(3:2:111)

I = fcos z½Sið2 zÞ − 2CiðzÞ − sin z ½Cið2 zÞ − 2CiðzÞg and for ν = 3/2 rffiffiffiffiffiffi        ∂ Hν 2 z z 1 z 2 ð1 − cos zÞ ln − + = +γ + ∂ν ν = 3=2 πz 2 2 z 2 z  cos z + sin z + ½Cið2 zÞ − 2CiðzÞ z    sin z + − cos z Sið2 zÞ − 2SiðzÞ z

(3:2:112)

Thus, by using (3.2.108), (3.2.111) and (3.2.112) it is possible, consecutively to determine derivatives with respect to the order for ν = n + 1/2, n = 1,2,3, . . . Similarly, as the Bessel function in (3.2.95), the Struve functions Hν(t) and Lν(t) are interrelated Lν ðtÞ = − i e− i π ν=2 H ν ðitÞ

(3:2:113)

Differentiation of (3.2.113) with respect to the order ν gives ∂ Lν ðtÞ πi ∂ H ν ðitÞ = − Lν ðtÞ − i e− i π ν=2 ∂ν 2 ∂ν and the recurrence relation is       ∂ Lν − 1 ðzÞ ∂ Lν + 1 ðzÞ 2ν ∂ Lν ðzÞ 2 − = + Lν ðzÞ ∂ν ∂ν z ∂ν z z ν     z 3 2  p ffiffiffi + ln −ψ ν+ 2 2 π Γ ν + 32

(3:2:114)

(3:2:115)

Using (3.2.114) and (3.2.115) we have   ∂ Lν 1 = pffiffiffiffiffiffiffiffiffi f e−z ½Eið2zÞ − 2EiðzÞ + ez ½E1 ð2 zÞ − 2E1 ðzÞg ∂ν ν = − 1=2 2π z ðz ðex − 1Þ

EiðzÞ = γ + ln z + ∞ ð

E1 ðzÞ = z

dx x

0

e−x

dx = − Eið−zÞ = Γð0, zÞ x

(3:2:116)

101

3.3 Derivatives with Respect to the Order v

and rffiffiffiffiffiffi   ∂ Lν 2 I =− ∂ν ν = 1=2 πz

h  i  z e−z ez ½E1 ð2 zÞ − 2E1 ðzÞ I = ln ½Eið2zÞ + 2EiðzÞ + +γ − 2 2 2

(3:2:117)

Using the general formula for derivatives of the modified Struve functions       ∂ Lν ðzÞ z 1 = ln Lν ðzÞ −ψ ν+ ∂ν 2 2  ν ð1 2 z2 1 ν − 1=2 ð1 − t2 Þ + pffiffiffi  lnð1 − t2 Þ sinhðz tÞ dt ; Re ν > − 1 2 πΓ ν + 2 0

(3:2:118) and (3.2.117), it possible, as a by-product, to evaluate the following integral ð1 lnð1 − t2 Þ sinhðz tÞ dt = −

1 h z i ln + γ cosh z z 2

(3:2:119)

0

e−z ez ½Eið2 zÞ − 2EiðzÞ − ½E1 ð2zÞ − 2E1 ðzÞ + 2 2

3.3 Derivatives with Respect to the Order ν of the Bessel and Related Functions Based on Integral Representations Using the Leibniz rule of differentiating integrals, almost any integral with integrand containing Bessel or related functions is a starting point in derivation of derivatives with respect to the order. Evidently, the same happens with the integral representations of the Bessel and related functions. One of well-known examples is ∂ Yν ðzÞ ∂ Jν ðzÞ 4 − Yν ðzÞ =− Jν ðzÞ ∂ν ∂ν π

∞ ð

e− 2νt K0 ð2 z sinh tÞ dt (3:3:1)

0

Re z > 0 The expression (3.3.1) was derived similar to the Nicholson integral [7] 8 ½Jν ðzÞ + ½Yν ðzÞ = 2 π 2

∞ ð

2

K0 ð2 z sinh tÞ coshð2ν tÞ dt 0

;

Re z > 0

(3:3:2)

102

3 Differentiation and Integration with Respect to the Order of the Bessel

Dunster [48], by solving the Bessel differential equation with respect to the order by the method of variation of parameters showed that for ν > 0 and Re z > 0 it is possible to derive similar expressions ∂ Jν ðzÞ = π ν Yν ðzÞ ∂ν ∂ Yν ðzÞ = π νJν ðzÞ ∂ν

ðz

½Jν ðtÞ2 dt + π νJν ðzÞ t

Jν ðtÞYν ðtÞ dt t

(3:3:3)

z

0

ðz

∞ ð

½Yν ðtÞ2 dt − π ν Yν ðzÞ t

∞ ð

Jν ðtÞYν ðtÞ π dt − Jν ðzÞ t 2

(3:3:4)

z

0

Thus, derivatives with respect to the order can be expressed in a closed form by using integral representations of Bessel and related functions. Usually, differentiations should be performed in three places in integral representations, one place is the differentiation under integral sign (differentiation of integrand). In some cases, the differentiated integral can be evaluated for particular values of v and the final result can be expressed in terms of elementary or special functions. Oberhettinger [56] applied the following integral representation of the modified Bessel function of the second kind to obtain derivatives for v = ± 1/2 pffiffiffi z ν ∞ ð π Kν ðzÞ =  2 1 e− z t ðt2 − 1Þν − 1=2 dt Γ ν+ 2 (3:3:5) 1 1 Re z > 0 ; Re ν > − 2 Differentiation of (3.3.5) gives     pffiffiffi z ν ∞ ð π ∂ Kν ðzÞ z 1  2 1 e−z t ðt2 − 1Þν − 1=2 dt = ln −ψ ν+ ∂ν 2 2 Γ ν+ 2 1

pffiffiffi z ν ∞ ð π +  2 1 e−z t lnðt2 − 1Þ ðt2 − 1Þν − 1=2 dt Γ ν+ 2

(3:3:6)

1

or in the equivalent form pffiffiffi z ν ∞     ð π ∂ Kν ðzÞ z 1 Kν ðzÞ +  2 1 e−z t lnðt2 − 1Þ ðt2 − 1Þν − 1=2 dt = ln −ψ ν+ ∂ν 2 2 Γ ν+ 2 1

(3:3:7) Introducing v = 1/2 into (3.3.7) we have

3.3 Derivatives with Respect to the Order v

103

rffiffiffiffiffi h ffi     z i rffiffiffiffiffi ∂ Kν ðzÞ π −z πz 1 = e γ + ln + I z, ∂ν ν = 1=2 2z 2 2 2   ∞ ð 1 I z, = e−z t lnðt2 − 1Þ dt 2

(3:3:8)

1

rffiffiffiffiffi π −z K1=2 ðzÞ = e 2z

;

ψð1Þ = − γ

Oberhettinger showed that this integral can be integrated by parts and the final result of integration is ∞  −z t    ∞ ð ð 1 e dt = e−z t lnðt2 − 1Þ dt = lnðt2 − 1Þ d − I z, z 2 1

1

z i 1 e−z h = − ez Eið− 2 zÞ − γ + ln z z 2 Combining (3.3.8) with (3.3.9) gives the same expression as in (3.2.103) rffiffiffiffiffi   ∂ Kν ðzÞ π z =− e Eið− 2zÞ ∂ν ν = 1=2 2z

(3:3:9)

(3:3:10)

and using Kν ðzÞ = K −ν ðzÞ ∂ K−ν ðzÞ ∂ Kν ðzÞ =− ∂ν ∂ν the derivative for v = −1/2 is determined as rffiffiffiffiffi   ∂ Kν ðzÞ π z = e Eið− 2zÞ ∂ν ν = − 1=2 2z

(3:3:11)

(3:3:12)

Derivatives with respect to the order for other Bessel functions with v = ± 1/2 (see, for example, expressions (3.2.106) or (3.2.117)) were derived by Oberhettinger. He differentiated the Bessel functions’ relations when argument z is reduced to the principal branch |z| < π (for details see [56]). The integral representation of the Bessel and related functions (denoted here as Fν(z)) can be formally written in the following form z ν ðb Fν ðzÞ = A  2 1 Gðt, νÞ dt Γ ν+ 2 a

(3:3:13)

104

3 Differentiation and Integration with Respect to the Order of the Bessel

where A is a numerical constant and the limits of integration a and b are constants or infinity. The general result for derivatives with respect to the order ν based on integral representations is available from a direct differentiation of (3.3.13) z ν     ∂ Fν ðzÞ z 1 Fν ðzÞ + A  2 1 Iðt, νÞ = ln −ψ ν+ ∂ν 2 2 Γ ν+ 2 (3:3:14) ðb ∂ Gðt, νÞ Iðt, νÞ = dt ∂ν a

Thus, derivatives of different Bessel functions have the same form, but they contain different expressions for integrals I(t,ν). In mathematical manipulations, the formulas for derivatives given in (3.3.14) are superior than those expressed by infinite or finite series. They include only the function under consideration and one integral that should be evaluated. In some cases, these integrals can be presented in closed form, as demonstrated by Oberhettinger for v = ± 1/2 [56]. Usually, these integrals can be evaluated by rather simple numerical methods. For the first time, in a systematic way, the expression (3.3.14) is applied here to obtain derivatives with respect to the order by using different integral representations of the Bessel and related functions given in Section 2.12. Let us start with the Bessel function of the first kind which has the following integral representation (2.12.13)  ν ð1 2 z2 ð1 − t2 Þν − 1=2 cosðz tÞ dt Jν ðzÞ = pffiffiffi  π Γ ν + 21 0

Re z > 0

; Re ν > −

(3:3:15)

1 2

From (3.3.14) we have  ν     2 z ∂ Jν ðzÞ z 1 Jν ðzÞ + pffiffiffi 2 1 Iðt, νÞ = ln −ψ ν+ ∂ν 2 2 πΓ ν+ 2 ð1

(3:3:16) 2 ν − 1=2

ð1 − t Þ

Iðt, νÞ =

lnð1 − t Þ cosðz tÞ dt 2

0

For some values of ν the above integral can be expressed in terms of special functions. For ν = 0, we have

3.3 Derivatives with Respect to the Order v

105

      ∂ Jν ðzÞ z 1 2 J0 ðzÞ + Iðt, 0Þ = ln −ψ ∂ν ν = 0 2 2 π ð1 Iðt, 0Þ = 0

lnð1 − t2 Þ cosðz tÞ pffiffiffiffiffiffiffiffiffiffi dt 1 − t2

(3:3:17)

  1 ψ = −γ −2 ln 2 2 This derivative with respect to the order is given in (3.2.20) and therefore by comparing it with (3.3.17) as a by-product the integral in (3.3.17) can be evaluated ð1 0

lnð1 − t2 Þ cosðz tÞ π pffiffiffiffiffiffiffiffiffiffi dt = fπ Y0 ðzÞ − 2½lnð2 zÞ + γ J0 ðzÞg 2 4 1−t

(3:3:18)

Introducing ν = 1 into (3.3.16) gives       ∂ Jν ðzÞ z 3 2z J1 ðzÞ + = ln −ψ Iðt, 1Þ ∂ν ν = 1 2 2 π ð1 pffiffiffiffiffiffiffiffiffiffi 1 − t2 lnð1 − t2 Þ cosðz tÞ dt Iðt, 1Þ =

(3:3:19)

0

and by comparing (3.2.22) with (3.3.319) the logarithmic integral is   ð1 pffiffiffiffiffiffiffiffiffiffi π 1 π 2 2 1 − t lnð1 − t Þ cosðz tÞ dt = J0 ðzÞ + Y1 ðzÞ − ðlnð2zÞ − 2 + γÞ J1 ðzÞ 2z z 2 0

(3:3:20) Similarly, for ν = 1/2 rffiffiffiffiffiffih  i rffiffiffiffiffi     ∂ Jν ðzÞ 2 z 2z 1 ln I t, = + γ sin z + ∂ν ν = 1=2 πz 2 π 2   ð1 1 = lnð1 − t2 Þ cosðz tÞ dt I t, 2

(3:3:21)

0

with help of (3.2.66) we have ð1

h z i 1 lnð1 − t Þ cosðz tÞ dt = ½cos z Cið2 zÞ + sin z Sið2 zÞ − sin z ln +γ z 2 2

0

(3:3:22) If the integrand in (3.3.14) includes trigonometric functions

106

3 Differentiation and Integration with Respect to the Order of the Bessel

z ν ðπ Jν ðzÞ = pffiffiffi 2 cosðz cos tÞðsin tÞ2 ν dt π Γ ν + 21

(3:3:23)

0

1 Re ν > − 2 then the differentiation of this expression with respect to ν under the integral sign gives  ν     2 z2 ∂ Jν ðzÞ z 1 Iðt, νÞ Jν ðzÞ + pffiffiffi  = ln −ψ ν+ ∂ν 2 2 π Γ ν + 21 (3:3:24) ðπ lnðsin tÞ cosðz cos tÞ ðsin tÞ2 ν dt

Iðt, νÞ = 0

and from (3.3.16) and (3.3.24) we have the equality of integrals ð1 ð1 − t2 Þ

ν − 1=2

ðπ lnð1 − t2 Þ cosðz tÞ dx =

0

lnðsin tÞ cosðz cos tÞðsin tÞ2 ν dt

(3:3:25)

0

If integral representations are in the form given in (3.3.14), but they have different integrands and limits of integration, then the equality of integrals in general case can be written as ðb A1 a

∂ G1 ðt, νÞ dt = A2 ∂ν

ðd c

∂ G2 ðt, νÞ dt ∂ν

The first integral representations of the Bessel functions was given by Schlaefli [7] Jν ðzÞ =

1 π

ðπ cosðz sin t −ν tÞ dt − 0

sinðπ νÞ π

∞ ð

e−z sinh t −νt dt

0

(3:3:26)

Re z > 0 and by a direct differentiation of it we have ∂ Jν ðzÞ 1 = ∂ν π

ðπ 0

1 t sinðz sin t −ν tÞdt + π

∞ ð

e −z sinh t −νt ½t sinðπ νÞ − π cosðπ νÞ dt

0

(3:3:27) For ν = 0 it reduces to ∞   ð ðπ ∂ Jν ðzÞ 1 = t sinðz sin tÞdt − e−z sinh t dt ∂ν ν = 0 π 0

0

(3:3:28)

3.3 Derivatives with Respect to the Order v

107

but π 1 Y0 ðzÞ = 2 2

∞ ð

ðπ sinðz sin tÞdt − 0

e−z sinh t dt

(3:3:29)

0

and 1 2

ðπ sinðz sin tÞ dt = 0

1 π

ðπ t sinðz sin tÞ dt

(3:3:30)

0

and therefore the expected result is derived   ∂ Jν ðzÞ π = Y0 ðzÞ ∂ν ν = 0 2

(3:3:31)

Differentiation of the Schlaefli integral representation for the Bessel function of the second kind 1 Yν ðzÞ = π

ðπ 0

1 sinðz sin t −νtÞdt − π

∞ ð

e−z sinh t ½eν t + cosðπνÞ e−ν t  dt

0

(3:3:32)

Re z > 0 gives ∂ Yν ðzÞ 1 =− ∂ν π 1 − π

ðπ t cosðz sin t −νtÞdt 0 ∞ ð

(3:3:33) e

−z sinh t

νt

ft ½e − cosðπνÞ e

−ν t

−ν t

 − π sinðπνÞ e

g dt

0

When introducing ν = 0, the second integral in (3.3.33) vanishes, combines with (3.3.26) and we have   ðπ ∂ Yν ðzÞ 1 =− t cosðz sin tÞdt ∂ν ν = 0 π ðπ

0

2 t cosðz sin tÞdt π2 0   ∂ Yν ðzÞ π = − J0 ðzÞ ∂ν ν = 0 2 J0 ðzÞ =

and the equivalent to (3.3.30) expression for the equality of integrals

(3:3:34)

108

3 Differentiation and Integration with Respect to the Order of the Bessel

1 2

ðπ cosðz sin tÞ dt = 0

ðπ

1 π

t cosðz sin tÞ dt

(3:3:35)

0

Integral representation of the modified Bessel function of the first kind also includes two integrals 1 Iν ðzÞ = π Re z > 0

ðπ z cos t

e 0

sinðπ νÞ cosðν tÞdt − π

∞ ð

e−z cosh t −νt dt

0

(3:3:36)

; Re ν > 0

and therefore ∂ Iν ðzÞ 1 =− ∂ν π

ðπ t ez cos t sinðν tÞdt + 0

1 π

∞ ð

e−z cosh t −ν t ½t sinðπνÞ − π cosðπνÞ dt

0

(3:3:37) From the integral representation of the modified Bessel function of the second kind ∞ ð

Kν ðzÞ =

e−z cosh t coshð νtÞ dt

;

Re z > 0

(3:3:38)

0

it is possible to obtain ∂ Kν ðzÞ = ∂ν

∞ ð

t e−z cosh t sinhð νtÞ dt

(3:3:39)

0

The derivatives of the Bessel functions of the first and second kind derived above are useful in the case of the Anger and Weber functions J ν ðzÞ = Jν ðzÞ +

∞ ð

sinðπ νÞ π

Eν ðzÞ = − Yν ðzÞ −

1 π

∞ ð

e−z sinh t −ν t d t

(3:3:40)

0

e−z sinh t ½eν t + e−ν t cosðπ νÞ d t

(3:3:41)

0

and therefore ∂ J ν ðzÞ ∂ Jν ðzÞ 1 = + ∂ν ∂ν π

∞ ð

0

e−z sinh t −ν t ½π cosðπ νÞ − t sinðπ νÞ dt

(3:3:42)

109

3.3 Derivatives with Respect to the Order v

∂ Eν ðzÞ ∂ Yν ðzÞ 1 =− − ∂ν ∂ν π

∞ ð

e−z sinh t f t ½eν t − cosðπ νÞ e−ν t  − π sinðπ νÞ e−ν t gd t

0

(3:3:43) In particular case of ν = 0, we have ∞   ð ∂ J ν ðzÞ π = Y0 ðzÞ + e−z sinh t dt ∂ν 2 ν =0

(3:3:44)

0

but the Anger function is defined by the integral J ν ðzÞ =

ðπ 1 cosðν t −z sin tÞdt π

(3:3:45)

0

and its differentiation yields ∂ J ν ðzÞ 1 =− ∂ν π

ðπ t sinðνt −z sin tÞ d t

(3:3:46)

0

therefore for ν = 0, the expression in (3.3.46) becomes   ðπ ∂ J ν ðzÞ 1 = t sinðz sin tÞ dt ∂ν π ν =0

(3:3:47)

0

Comparing (3.3.47) with (3.3.32.) gives Y0 ðzÞ = −

2 π

∞ ð

e−z sinh t d t −

0

2 π2

ðπ t sinðz sin tÞ dt

(3:3:48)

0

and since [18]   ∂ J ν ðzÞ π = H0 ðzÞ ∂ν 2 ν =0

(3:3:49)

we have 2 H 0 ðzÞ = Y0 ðzÞ + π

∞ ð

0

e−z sinh t dt

(3:3:50)

110

3 Differentiation and Integration with Respect to the Order of the Bessel

and 2 H 0 ðzÞ = 2 π

ðπ t sinðz sin tÞ dt

(3:3:51)

0

Similarly, it is possible to obtain     ∂ Eν ðzÞ ∂ Yν ðzÞ π =− = J0 ðzÞ ∂ν ∂ν 2 ν =0 ν =0

(3:3:52)

From Eν ðzÞ =

ðπ 1 sinðνt −z sin tÞ dt π

(3:3:53)

0

by differentiation of (3.3.53) we have ∂ Eν ðzÞ 1 = ∂ν π

ðπ t cosðν t −z sin tÞ dt

(3:3:54)

0

and for ν = 0, once again the result given in (3.3.52) is reached. If a number of evaluated integrals is considered, the Schlaefli integral representations are less convenient to obtain derivatives of the Bessel and related functions with respect to the order than those based on (3.3.14) expression. Using integral representations given in Section 2.12, it is possible to present a short list of first derivatives. In all cases of derived formulas, conditions Re z > 0 and Re ν > − 1/2 are imposed. The first expression is a different form of derivative with respect to the order of the Bessel function of the first kind     ∂ Jν ðzÞ z 1 Jν ðzÞ = ln −ψ ν+ ∂ν 2 2 (3:3:55)  z ν ð1 ν pffiffiffiffiffiffiffiffi t ln t 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðz 1 − tÞ dt + pffiffiffi π Γ ν + 21 tð1 − tÞ 0

For ν = 0 and ν = 1 it becomes that   ∂ Jν ðzÞ = ½ln z + γ + ln 2 J0 ðzÞ ∂ν ν = 0 1 + π

ð1 0

pffiffiffiffiffiffiffiffi ln t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðz 1 − tÞ dt tð1 − tÞ

(3:3:56)

3.3 Derivatives with Respect to the Order v

  ð1 rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ∂ Jν ðzÞ z t ln t cosðz 1 − tÞ dt = ½ln z + γ + ln 2 − 2 J1 ðzÞ + ∂ν ν = 1 π 1−t

111

(3:3:57)

0

These integrals can be evaluated by using (3.3.31) and (3.2.24). In case of the modified Bessel functions, we have     ∂ Iν ðzÞ z 1 Iν ðzÞ = ln −ψ ν+ ∂ν 2 2  ν ðπ 2 z + pffiffiffi 2 1 lnðsin tÞðsin tÞ2 ν coshðz tÞ dt πΓ ν+ 2

(3:3:58)

0

    ∂ Kν ðzÞ z 1 Kν ðzÞ = ln −ψ ν+ ∂ν 2 2 pffiffiffi  ν ∞ ð 2 π z2  + e−z cosh t lnðsinh tÞðsinh tÞ2 ν dt Γ ν + 21

(3:3:59)

0

Also in this case, integrals in (3.3.58) and (3.3.59) can be expressed in a closed form for ν = 0 and ν = 1. Derivatives with respect to the order of the Struve functions are     ∂ H ν ðzÞ z 1 H ν ðzÞ = ln −ψ ν+ ∂ν 2 2 (3:3:60)  ν ð1 2 z2 ν − 1=2 2 2 ð1 − t Þ + pffiffiffi  lnð1 − t Þ sinðz tÞ dt π Γ ν + 21 0

    ∂ H ν ðzÞ z 1 H ν ðzÞ = ln −ψ ν+ ∂ν 2 2 π=2  ν ð 4 z2 sinðz cos tÞ lnðsin tÞ ðsin tÞ2 ν dt + pffiffiffi  π Γ ν + 21

(3:3:61)

0

    ∂ Lν ðzÞ z 1 Lν ðzÞ = ln −ψ ν+ ∂ν 2 2 π=2  ν ð 4 z2  sinhðz cos tÞ lnðsin tÞ ðsin tÞ2 ν dt + pffiffiffi π Γ ν + 21

(3:3:62)

0

For ν = 0 and ν = 1, by using different integral representations of the Struve functions, it is possible to obtain [5]:

112

3 Differentiation and Integration with Respect to the Order of the Bessel

  h z i ∂ H ν ðzÞ π = − J0 ðzÞ + ln + γ + 2 ln 2 ½H 0 ðzÞ − Y0 ðzÞ ∂ν 2 2 ν =0 ∞ ð 2 lnð1 + t2 Þ + e−z t pffiffiffiffiffiffiffiffiffiffi dt π 1 + t2

(3:3:63)

0

  h z i ∂ H ν ðzÞ = ln + γ + 2 ln 2 − 2 ½H 1 ðzÞ − Y1 ðzÞ ∂ν 2 ν =1 ∞ ð pffiffiffiffiffiffiffiffiffiffi 2z 1 π + e−z t 1 + t2 lnð1 + t2 Þ dt − Y0 ðzÞ − J1 ðzÞ π z 2

(3:3:64)

0

  h z i ∂ Lν ðzÞ π = − J0 ðzÞ + ln + γ + 2 ln 2 L0 ðzÞ ∂ν 2 2 ν =0 2 + π

ð1 0

lnð1 − t2 Þ pffiffiffiffiffiffiffiffiffiffi sinhðz tÞ dt 1 − t2

(3:3:65)

  h z i ∂ Lν ðzÞ = ln + γ + 2 ln 2 − 2 L1 ðzÞ ∂ν 2 ν =1 2z + π

ð1 e

−z t

pffiffiffiffiffiffiffiffiffiffi 1 − t2 lnð1 − t2 Þ sinhðz tÞ dt

(3:3:66)

0

The asymptotic behaviour of integrals given in the above expressions is known for small and large values of argument z [5]. The Struve functions can be expressed also with the help of the Bessel functions and therefore it is possible to obtain from (2.12.60) and (2.12.64)     ∂ H ν ðzÞ ∂ Yν ðzÞ z 1 H ν ðzÞ = + ln −ψ ν+ ∂ν ∂ν 2 2 ∞  ν (3:3:67) ð 2 z2 ν − 1=2 −z t 2 2 e ð1 + t Þ + pffiffiffi  lnð1 + t Þ dt π Γ ν + 21 0

    ∂ Lν ðzÞ ∂ Iν ðzÞ z 1 Lν ðzÞ = − ln −ψ ν+ ∂ν ∂ν 2 2  ν ð1 2 z ν − 1=2 − pffiffiffi 2 1 e −z t ð1 − t2 Þ lnð1 − t2 Þ dt πΓ ν+ 2 0

These integrals for ν = 0 and ν = 1/2, can be determined because

(3:3:68)

3.3 Derivatives with Respect to the Order v



∂ H ν ðzÞ ∂ν

 2 1 π 1 32 z  = − J0 ðzÞ + 2 G24  4  1=2, 1, 2 πz ν=0



1

113

! ;

1, 1=2

Re z ≥ 0

(3:3:69)

and 

∂ H ν ðzÞ ∂ν

rffiffiffiffiffiffin z 2 γ + ln = + cos z ½Cið2zÞ − 2CiðzÞ πz 2 ν = 1=2



+ sin z ½Sið2zÞ − 2SiðzÞg

(3:3:70)

Thus, we have ∞ ð

0

e −z t lnð1 + t2 Þ π pffiffiffiffiffiffiffiffiffiffi dt = − ðln z + γ + ln 2Þ H 0 ðzÞ 2 2 1+t  ! 2 1 1 1 32 z  + G  2 π z 24 4  1=2, 1, 1, 1=2

(3:3:71) ;

Re z ≥ 0

and ∞ ð

e −z t lnð1 + t2 Þ dt = ½π − 2SiðzÞ sin z + ½ln z − ln 2 + γ − 2CiðzÞ cos z

(3:3:72)

0

In the case of the modified Struve function for ν = 0 and ν = 1/2, from !    2 1 1 ∂ Lν ðzÞ 1 42 z  = K0 ðzÞ − 2 G24 4  1=2, 1, 1, 1=2 ∂ν ν = 0 πz !  1, 1, 1=4, 3=4 2 42 z2  = − K0 ðzÞ − G46 ; Re z ≥ 0 4  1=2, 1=2, 1, 1, 1=4, 3=4 z   ∂ Iν ðzÞ = − K0 ðzÞ ∂ν ν = 0

(3:3:73)

(3:3:74)

it is possible to derive that   z i ∂ Lν ðzÞ 1 n h = − pffiffiffiffiffiffiffiffiffi 2 γ + ln + e −z ½Eið2zÞ − 2EiðzÞ ∂ν ν = 1=2 2 2πz + ez ½Eið− 2zÞ − 2Eið− zÞg

  ∂ Iν ðzÞ 1 = pffiffiffiffiffiffiffiffiffi ½ez Eið− 2zÞ − e−z Eið2zÞ ∂ν ν = 1=2 2π z

(3:3:75) (3:3:76)

114

3 Differentiation and Integration with Respect to the Order of the Bessel

and ð1 0

 ! 1, 1, 1=4, 3=4 e−z t lnð1 − t2 Þ π 42 z2  π pffiffiffiffiffiffiffiffiffiffi = G46 − ½ln z + γ + ln 2 L0 ðzÞ  4  1=2, 1=2, 1, 1, 1=4, 3=4 z 2 1 − t2 (3:3:77)

ð1

e−z t lnð1 − t2 Þ dt = −

i 1 h z ln + γ + ez Eið−zÞ − e−z EiðzÞ z 2

(3:3:78)

0

In some cases, differentiation with respect to the order is easier to perform if in the integral representation of the Bessel and related functions the order ν is absent in the pre-integral factor. Such situation appears in the Mehler–Sonine formulas [7] 2 Jν ðzÞ = π Yν ðzÞ = −

∞ ð

 π ν sin z cosh t − coshðν tÞ dt 2

;

− 1 < Re ν < 1

(3:3:79)

0

2 π

∞ ð

 π ν cos z cosh t − coshðνtÞ dt 2

;

− 1 < Re ν < 1

(3:3:80)

0

In this case, these integral representations are valid for a limited range of the order ν. Differentiation of (3.3.79) gives ∂ Jν ðzÞ ∂ν ∞  ð   2t π ν π ν = sin z cosh t − sinhðν tÞ − cos z cosh t − coshðνtÞ dt π 2 2

(3:3:81)

0

and for ν = 0, the expected result is achieved ∞   ð ∂ Jν ðzÞ π = Y0 ðzÞ = − cosðz cosh tÞ dt ∂ν 2 ν =0

(3:3:82)

0

Similarly from ∂ Yν ðzÞ ∂ν ∞  ð   2t π ν π ν =− cos z cosh t − sinhðν tÞ + sin z cosh t − coshðνtÞ dt π 2 2 0

(3:3:83)

3.3 Derivatives with Respect to the Order v

115

we have for ν = 0 ∞   ð ∂ Jν ðzÞ π = − J0 ðzÞ = − sinðz cosh tÞ dt ∂ν 2 ν =0

(3:3:84)

0

Integral representations of the modified Bessel functions of the second kind which are given in (2.12.45) – (2.12.55), have the order ν in one or two places only. The simplest case is ∞ ð

e−z cosh t coshð νtÞ dt

Kν ðzÞ =

(3:3:85)

0

and therefore ∂ Kν ðzÞ = ∂ν

∞ ð

t e−z cosh t sinhðν tÞ dt

(3:3:86)

0

For ν = 0, this expression shows that the first derivative of Kν(z) with the respect to the order ν is zero. In the case ν = 1, using (3.3.85) and (3.3.86), we have ∞ ð

K0 ðzÞ = z

t e−z cosh t sinh t dt

(3:3:87)

0

and for ν = 1/2 it is possible to deduce that rffiffiffiffiffi ∞   ð 2z −z t e dt t e−z cosh t sinh − Eið− 2zÞ = π 2

(3:3:88)

0

The Kelvin functions of the first kind and the order ν have the following integral representations of the Schlaefli type [6] ðπ pffiffiffi 1 berν ð 2 zÞ = ½cosðπ νÞ cosðz sin t −νtÞ coshðz sin tÞ π 0

− sinðπ νÞ sinðz sin t −νtÞ sinhðz sin tÞ dt ∞ ð sinðπ νÞ − e−ν t −z sinh t cosðz sinh t + π νÞ dt π 0

and the Kelvin functions of the second kind and the order ν are

(3:3:89)

116

3 Differentiation and Integration with Respect to the Order of the Bessel

ðπ pffiffiffi 1 beiν ð 2 zÞ = ½cosðπ νÞ sinðz sin t −ν tÞ sinhðz sin tÞ π 0

+ sinðπ νÞ cosðz sin t −νtÞ coshðz sin tÞ dt ∞ ð sinðπ νÞ − e−ν t −z sinh t sinðz sinh t + π νÞ dt π

(3:3:90)

0

Direct differentiation of these formulas gives pffiffiffi pffiffiffi ∂ berν ð 2 zÞ = − π beiν ð 2 zÞ ∂ν ðπ 1 + t ½sinðπ νÞ cosðz sin t −νtÞ sinhðz sin tÞ π 0

− cosðπ νÞ sinðz sin t −νtÞ coshðz sin tÞ dt ∞ ð 1 e−ν t −z sinh t ½t sinðπ νÞ − π cosðπ νÞ cosðz sinh t + π νÞ dt + π 0

(3:3:91) and pffiffiffi pffiffiffi ∂ beiν ð 2 zÞ = − π berν ð 2 zÞ ∂ν ðπ 1 + t ½sinðπ νÞ cosðz sin t −νtÞ coshðz sin tÞ π 0

− cosðπ νÞ cosðz sin t −νtÞ sinhðz sin tÞ dt +

1 π

∞ ð

e−ν t −z sinh t ½t sinðπ νÞ − π cosðπ νÞ sinðz sinh t + π νÞ dt

0

(3:3:92) These formulas have a much simpler form if ν = 0 pffiffiffi   ðπ pffiffiffi ∂ berν ð 2 zÞ 1 = − π beið 2 zÞ − t sinðz sin tÞ coshðz sin tÞ dt ∂ν π ν =0 0

∞ ð

−

e 0

−z sinh t

cosðz sinh tÞ dt

(3:3:93)

3.3 Derivatives with Respect to the Order v

117

and pffiffiffi   ðπ pffiffiffi ∂ beiν ð 2 zÞ 1 = − π berð 2 zÞ − t cosðz sin tÞ sinhðz sin tÞ dt ∂ν π ν =0 0

∞ ð

+

e

−z sinh t

(3:3:94)

sinðz sinh tÞ dt

0

In the case of the modified Kelvin functions kerν(z), keiν(z) and herν(z), heiν(z), the first step is to derive the integral representations from their definitions kerν ðzÞ + i keiν ðzÞ = e− π i=2 Kν ðzeπ i=4 Þ kerν ðzÞ − i keiν ðzÞ = eπ i=2 Kν ðze− π i=4 Þ

(3:3:95)

1±i e ± π i=4 = pffiffiffi 2 and from (2.12.45) we have ∞ ð

Kν ðzÞ =

e−z cosh t coshð νtÞ dt

;

Re z > 0

(3:3:96)

0

Combining (3.3.95) with (3.3.96), after many but elementary steps, the integral representations of the modified Kelvin functions are ∞ ð  pffiffiffi π ν coshðνtÞdt kerν ð 2 zÞ = e−z cosh t cos z cosh t + 2

(3:3:97)

0

and ∞ ð  pffiffiffi π ν coshðνtÞdt keiν ð 2 zÞ = − e−z cosh t sin z cosh t + 2

(3:3:98)

0

Thus, the derivatives of the modified Kelvin functions are available by a direct differentiation of (3.3.97) and (3.3.98) with respect to the order ν ∞ pffiffiffi ð h  ∂ kerν ð 2 zÞ π ν = e−z cosh t t cos z cosh t + sinhðνtÞ ∂ν 2 0

π



i π ν sin z cosh t + coshðνtÞ dt − 2 2

(3:3:99)

118

3 Differentiation and Integration with Respect to the Order of the Bessel

and ∞ pffiffiffi ð h  ∂ keiν ð 2 zÞ π ν = − e−z cosh t t sin z cosh t + sinhðν tÞ ∂ν 2 0π  i π ν cos z cosh t + coshðνtÞ dt + 2 2

(3:3:100)

For ν = 0, both the expressions take the form ∞ pffiffiffi   ð ∂ kerν ð 2 zÞ π =− e−z cosh t sinðz cosh tÞ dt ∂ν 2 ν =0

(3:3:101)

0

and ∞ pffiffiffi   ð ∂ keiν ð 2 zÞ π =− e−z cosh t cosðz cosh tÞ dt ∂ν 2 ν =0

(3:3:102)

0

Derivatives with respect to the order of the Lommel functions can be derived for the symmetrical case ν = μ with the help of derivatives of the Bessel or the Struve functions. However, if μ = 0 and μ = ± 1, the integral representations of the Lommel functions can be directly used. From Section 2.12. we have s − 1, ν ðzÞ = −

1 ν sinðπ νÞ

ðπ cosðz sin tÞ cosðν tÞ dt 0

s1, ν ðzÞ = 1 + ν s − 1, ν ðzÞ 2

s0, ν ðzÞ =

1 sinðπ νÞ ∞ ð

S0, ν ðzÞ =

(3:3:103) (3:3:104)

ðπ sinðz sin tÞ cosðν tÞ dt

(3:3:105)

0

e−z sinh t coshðνtÞ dt

;

Re z > 0

(3:3:106)

0 ∞ ð

S1, ν ðzÞ = z

e−z sinh t coshðνtÞ cosh t dt

;

Re z > 0

(3:3:107)

0

S−1, ν ðzÞ =

1 ½S1, ν ðzÞ − 1 ν2

(3:3:108)

119

3.4 Higher Order Derivatives of the Bessel and Related Functions

and therefore ( ) ðπ ∂ s−1, ν ðzÞ 1 = ½1 − π ν cotðπ νÞ s−1, ν ðzÞ − cscðπ νÞ t cosðz sin tÞ sinðνtÞ dt ∂ν ν 0

(3:3:109)   ∂ s1, ν ðzÞ ∂ s−1, ν ðzÞ = ν 2s−1, ν ðzÞ + ν ∂ν ∂ν ( ) ðπ ∂ s0, ν ðzÞ 1 =− ½π cotðπ νÞ s0, ν ðzÞ + t sinðz sin tÞ sinðνtÞ dt ∂ν sinðπ νÞ

(3:3:110)

(3:3:111)

0

∂ S0, ν ðzÞ = ∂ν ∂ S1, ν ðzÞ = ∂ν

∞ ð

0 ∞ ð

t e−z sinh t sinhðνtÞ dt

(3:3:112)

t e−z sinh t sinhðνtÞ cosh t dt

(3:3:113)

0

∂ S−1, ν ðzÞ 1 ∂S1, ν ðzÞ 2 = 2 − S−1, ν ðzÞ ∂ν ν ∂ν ν

(3:3:114)

The above expressions can be related to the Anger and the Weber functions J ν ðzÞ =

1 sinðπ νÞ ½s0, ν ðzÞ −ν s −1, ν ðzÞ π

Eν ðzÞ = −

1 + cosðπνÞ ν ½1 − cosðπνÞ s0, ν ðzÞ − s−1, ν ðzÞ π π

(3:3:115) (3:3:116)

and therefore we have   ∂ J ν ðzÞ 1 ∂ s0, ν ðzÞ ∂s−1, ν ðzÞ = π cotðπ νÞ J ν ðzÞ + sinðπ νÞ − − s−1, ν ðzÞ ∂ν π ∂ν ∂ν   ∂ Eν ðzÞ cosðπ νÞ − ðπ νÞ sinðπ νÞ − 1 s− 1, ν ðzÞ = − sinðπ νÞ s0, ν ðzÞ + ∂ν π     1 + cosðπ νÞ ∂ s0, ν ðzÞ ν cosðπ νÞ −ν ∂s−1, ν ðzÞ + + π ∂ν π ∂ν

(3:3:117)

(3:3:118)

3.4 Higher Order Derivatives of the Bessel and Related Functions with Respect to the Order ν Our knowledge about higher derivatives of the Bessel functions with respect to the order is very limited. Their significance considering properties of the Bessel functions

120

3 Differentiation and Integration with Respect to the Order of the Bessel

is still unclear. Derivations of them is not always easy, especially if performed with expressions having series. The first investigation devoted to numerical evaluation of the second derivative of the Bessel function Jν(z) with respect to the order for a wide range of orders and arguments is that of Airey [43] from 1935. For particular case of ν = 0, the second order derivatives were given by Luke [35] in his book dealing with integrals of the Bessel functions z k  2 

2 h  ∞ h  i  i X 2 ∂ Jν ðzÞ z π z 2 J ðzÞ J0 ðzÞ − 2 = π γ + ln + γ + ln Y0 ðzÞ − 2 k! k 6 k ∂ν2 2 2 ν =0 k =1 (3:4:1) and  k k z

2 h ∞ ð− 1Þ h z i z i2  X ∂ 2 Iν ðzÞ π 2 I ðzÞ I0 ðzÞ − 2 = − 2 γ + ln + γ + ln K0 ðzÞ − k 2 k! 6 k ∂ν2 2 2 k =1 (3:4:2) In 1977, Wienke [57] proposed to replace the power series representation of the derivatives with respect to the order of the Bessel function Jν(z) with the Neumann type expansion. ∞ z i X ∂ Jν ðzÞ h a2 k Jν + 2 k ðzÞ = γ + ln Jν ðzÞ − ∂ν 2 k =0   1 ð− 1Þk 2 k + ν a0 = γ + + ψðνÞ ; a2 k = k ν k+ν

(3:4:3)

ν ≠ − 1, − 2, − 3, . . . and ∞ z i2 h z i X ∂ 2 Jν ðzÞ h = γ + ln Jν ðzÞ − 2 γ + ln a2 k Jν + 2 k ðzÞ 2 ∂ν 2 2 k =0

−

∞ X ∂ a2 k

k =0

∂ν

∂ a0 1 = − 2 + ψ′ðνÞ ∂ν ν

Jν + 2 k ðzÞ +

∞ X ∞ X

a2 l a2 k Jν + 2 l + 2 k ðzÞ

(3:4:4)

l =0 k =0

;

∂ a2 k ð− 1Þk =− ∂ν ðν + kÞ2

; k>0

Wienke also gave the first derivatives with respect to the order when ν takes positive and negative integer values, they are presented in the Hansen Tables [15]. Evidently, by the direct differentiation of the series expansion of the Bessel function Jν(z) it is possible to obtain the following result

121

3.4 Higher Order Derivatives of the Bessel and Related Functions

z ∂ J ðzÞ h z i2 ∂ 2 Jν ðzÞ ν = 2 ln Jν ðzÞ − ln ∂ν2 2 ∂ν 2 n o ∞ z 2 k + ν k! ½ψðν + k + 1Þ2 − ψ′ðν + k + 1Þ X + ð− 1Þk Γðν + k + 1Þ 2 k =0

(3:4:5)

A more systematic investigation of the second order and third order derivatives of the Bessel functions has been recently performed by Brychkov. Only one of his expressions for ν = 0 is presented here  2   z  z 2 ∂ Jν ðzÞ 3 2 = π Y ðzÞ γ + ln − 0 3 F4 ð1, 1, ; 2, 2, 2, 2, ; − z Þ 2 4 ∂ν 2 2 ν=0  ! pffiffiffi  1=2, − 1=2 1 π2 π 31 2 − J0 ðzÞ + J0 ðzÞ G35 z   0, 0, 0, − 1=2, 0 4 2 Re z ≥ 0 (3:4:6) Other formulas for derivatives of the Bessel functions with integer orders are very long and complex. Since most of integral representations of the Bessel and related functions Fν(z) have the form given in (3.3.13) z ν ðb Fν ðzÞ = A  2 1 Gðt, νÞ dt Γ ν+ 2

(3:4:7)

a

the second derivative with respect to the order becomes      2 ∂2 Fν ðzÞ ′ ν + 1 Fν ðzÞ + ln z − ψ ν + 1 = − ψ Fν ðzÞ ∂ν2 2 2 2 ) z ν (     ðb ðb 2 z 1 ∂Gðt, νÞ ∂ Gðt, νÞ 2 2 ln +A  dt −ψ ν+ dt + 2 2 ∂ν ∂ν2 Γ ν + 21 a

a

(3:4:8) Thus, the formula for the second derivative with respect to the order includes only one additional mathematical operation, the differentiation of integrand of already known integral I(t,ν) ∂ Iðt, νÞ ∂ = ∂ν ∂ν

ðb a

∂ Gðt, νÞ dt = ∂ν

ðb a

∂ 2 Gðt, νÞ dt ∂ν2

If first derivative from (3.3.14) is preserved, (3.4.8) takes the form

(3:4:9)

122

3 Differentiation and Integration with Respect to the Order of the Bessel

      ∂2 Fν ðzÞ 1 z 1 ∂ Fν ðzÞ ð1Þ F = − ψ ν + ðzÞ + ln − ψ ν + ν 2 ∂ν 2 2 2 ∂ν z ν     ðb z 1 ∂Gðt, νÞ + A  2 1 ln −ψ ν+ dt 2 2 ∂ν Γ ν+ 2

(3:4:10)

a

z ν ðb 2 ∂ Gðt, νÞ dt + A  2 1 ∂ν2 Γ ν+ 2 a

For example in the case of the Bessel function Jν(z) we have       ∂2 Jν ðzÞ 1 z 1 2 ð1Þ J = − ψ ν + ðzÞ + ln Jν ðzÞ − ψ ν + ν ∂ν2 2 2 2  ν     ð1 4 z2 ln z2 − ψ ν + 21 ν − 1=2 pffiffiffi  + ð1 − t2 Þ lnð1 − t2 Þ cosðz tÞ dt π Γ ν + 21

(3:4:11)

0

 ν ð1 2 z ν − 1=2 + pffiffiffi 2 1 ð1 − t2 Þ ½lnð1 − t2 Þ2 cosðz tÞ dt πΓ ν+ 2 0

For ν = 0, the expression in (3.4.11) becomes  2   2 ∂ Jν ðzÞ π2 = − J0 ðzÞ + ðln z + γ + ln 2Þ 2 ∂ν2 ν =0 4 ðln z + γ + ln 2Þ + π 2 + π

ð1 0

ð1 0

lnð1 − t2 Þ pffiffiffiffiffiffiffiffiffiffi cosðz tÞ dt 1 − t2

(3:4:12)

2

½lnð1 − t2 Þ pffiffiffiffiffiffiffiffiffiffi cosðz tÞ dt 1 − t2

where the first integral is known ð1 0

lnð1 − t2 Þ cosðz tÞ π pffiffiffiffiffiffiffiffiffiffi dt = fπ Y0 ðzÞ − 2 ½lnð2 zÞ + γ J0 ðzÞg 2 4 1−t

For ν = 1/2, it follows from (3.4.11) that

(3:4:13)

3.4 Higher Order Derivatives of the Bessel and Related Functions

123

rffiffiffiffiffiffi   2  2 ∂ Jν ðzÞ 2 π2 − = sin z + ðln z + γ − ln 2Þ 6 ∂ν2 πz ν = 1=2 +

25=2

pffiffiffi ð1 z ðln z + γ − ln 2Þ lnð1 − t2 Þ cosðz tÞ dt π

(3:4:14)

0

pffiffiffi ð1 23=2 z 2 + ½lnð1 − t2 Þ cosðz tÞ dt π 0

where the first integral is also known ð1

h z i 1 lnð1 − t Þ cosðz tÞ dt = ½cos z Cið2zÞ + sin z Sið2zÞ − sin z ln +γ z 2 2

(3:4:15)

0

In the same way, using (3.4.10), it is possible to determine the second order derivatives for other Bessel functions when the first order derivatives are given as in Section 3.3. Third or higher order derivatives with respect to the order of the Bessel functions include repeated differentiation of the integral I(t,ν) and evidently they become longer and less convenient. As observed in (3.4.11), the only change in this expression or in those for higher derivatives will be powers of the logarithmic function in integrands of integrals. Integral representations of Bessel functions which have in their expressions less then three places of ν are also suitable for the determination of higher order derivatives with respect to the order. This occurs in the Schlaefli integral representations of the Bessel functions Jν ðzÞ =

1 π

ðπ cosðz sin t −ν tÞ dt − 0

sinðπ νÞ π

∞ ð

e−z sinh t −νt dt (3:4:16)

0

Re z > 0 and nth – derivative with respect to the order is ∂ n Jν ðzÞ 1 = ∂νn π

ðπ

1 − π

0

∂n ½cosðz sin t −ν tÞ dt ∂νn ∞ ð

0

(3:4:17) ∂n e−z sinh t n ½sinðπ νÞ e−νt  dt ∂ν

; n = 1, 2, 3, . . .

Before performing the differentiation, it is convenient to present the product of sine and exponential function by using the Euler complex form

124

3 Differentiation and Integration with Respect to the Order of the Bessel

ðπ

∂ n Jν ðzÞ 1 = ∂νn π

0

∂n ½cosðz sin t −ν tÞ dt ∂νn ∞ ð

1 − 2π i

0

(3:4:18) ∂n e−z sinh t n ½ e ðπ i − tÞν − e−ðπ i + tÞν  dt ∂ν

and therefore ( ð− 1Þn=2 ½1 + ð− 1Þn  t cosðz sin t −ν tÞ 2 0 ) ð− 1Þðn − 1Þ=2 ½1 − ð− 1Þn  sinðz sin t −ν tÞ dt + 2 ∞ ð   1 e−z sinh t −νt ðπ i − tÞn ei π ν − ð− 1Þn ðπ i + tÞn e−i π ν dt − 2π i ðπ

∂ n Jν ðzÞ 1 = ∂νn π

n

0

(3:4:19)

n = 0, 1, 2, 3, . . .

In the first integral, the sine function disappears for even order derivatives and the cosine function for odd order derivatives. Thus, from (3.4.19) first three order derivatives with respect to the order are ∂ Jν ðzÞ 1 = ∂ν π

ðπ t sinðz sin t −ν tÞ dt 0

1 + π

(3:4:20)

∞ ð

e

−z sinh t −νt

½t sinðπ νÞ − π cosðπ νÞ dt

0

∂ Jν ðzÞ 1 =− 2 ∂ν π 2

1 + π

ðπ t2 cosðz sin t −ν tÞ dt 0 ∞ ð

(3:4:21) e

−z sinh t −νt

½ðπ − t Þ sinðπ νÞ + 2 πt cosðπ νÞ dt 2

2

0

and ∂ 3 Jν ðzÞ 1 =− ∂ν3 π 1 + π

ðπ t3 sinðz sin t −ν tÞ dt 0 ∞ ð

(3:4:22) e

0

−z sinh t −νt

½ðt − 3 π tÞ sinðπ νÞ + ðπ − 3π t Þ cosðπ νÞ dt 3

2

3

2

125

3.4 Higher Order Derivatives of the Bessel and Related Functions

In the case of ν = 0, we will encounter three types of integrals ðπ I1 ðzÞ =

tn sinðz sin tÞ dt

;

n = 0, 1, 2, 3, . . .

0

ðπ I2 ðzÞ =

tn cosðz sin tÞ dt 0 ðπ

I3 ðzÞ =

(3:4:23)

tn e −z sinh t dt

0

From the Schlaefli integral representation of the Bessel function of second kind in (3.3.32) we have ∂ n Yν ðzÞ 1 = ∂νn π

ðπ t

n

ð− 1Þn=2 ½1 + ð− 1Þn  sinðz sin t −ν tÞ 2

0

 ð− 1Þðn + 1Þ=2 ½1 − ð− 1Þn  + cosðz sin t −ν tÞ dt 2 ∞ ð   1 e −z sinh t 2 tn eν t + e −ν t ½ðπ i − tÞn ei π ν + ð− 1Þn ðπ i + tÞn e −i π ν  dt − 2π 0

n = 0, 1, 2, 3, . . .

;

Re z > 0 (3:4:24)

which for ν = 0 it reduces to  n  ðπ n=2 ∂ Yν ðzÞ 1 ½1 + ð− 1Þn  n ð− 1Þ = t sinðz sin tÞ ∂νn π 2 ν =0 0

 ð− 1Þðn + 1Þ=2 ½1 − ð− 1Þn  + cosðz sin tÞ dt 2 ∞ ð 1 e −z sinh t f2tn + ½ðπ i − tÞn + ð− 1Þn ðπ i + tÞn g dt − 2π

(3:4:25)

0

The first three derivatives with respect to the order are ∂ Yν ðzÞ 1 =− ∂ν π 1 − π

ðπ t cosðz sin t −νtÞdt 0 ∞ ð

0

(3:4:26) (

)

e −z sinh t t eν t − ½t cosðπνÞ + π sinðπνÞ  e−ν t dt

126

3 Differentiation and Integration with Respect to the Order of the Bessel

∂ 2 Yν ðzÞ 1 =− ∂ν2 π 1 − π

ðπ t2 sinðz sin t −ν tÞ dt 0 ∞ ð

(3:4:27) e−z sinh t ft2 eν t + ½ðt2 − π2 Þ cosðπ νÞ − 2 tπ sinðπ νÞ e−ν t g dt

0

and ðπ

∂ 3 Yν ðzÞ 1 = ∂ ν3 π −

t3 cosðz sin t −ν tÞ dt 0

1 π

∞ ð

e−z sinh t ft3 eν t + ½ðπ3 − 3 π t2 Þ sinðπ νÞ

(3:4:28)

0

+ ð3 π2 t − t3 Þ cosðπ νÞ e −ν t gdt Derivatives with respect to the order of the modified Bessel function of the first kind are derived using (3.3.36) ( ) ðπ n=2 ∂ n Iν ðzÞ 1 ½1 + ð− 1Þn  ð− 1Þðn + 1Þ=2 ½1 − ð− 1Þn  n z cos t ð− 1Þ = t e cosðν tÞ + sinðν tÞ dt ∂νn π 2 2 0 ∞ ð ð− 1Þn e−z cosh t −ν t ½ðt − π iÞn ei π ν − ðt + π iÞn e −i π ν  dt − 2 πi n = 0, 1, 2, 3, . . .

;

0

Re z > 0

(3:4:29)

and they are for ν = 0  n  ðπ ∂ Iν ðzÞ ð− 1Þn=2 ½1 + ð− 1Þn  = tn ez cos t dt ∂νn 2π ν =0 n

ð− 1Þ − 2πi

∞ ð

0

(3:4:30)

e −z cosh t ½ðt − π iÞn − ðt + π iÞn  dt

0

where first integrals in (3.4.30) disappear for odd n. The first derivative with respect to the order of the modified Bessel function is given in (3.3.37), the second and third derivatives are presented below ∂ 2 Iν ðzÞ 1 =− 2 ∂ν π 1 + π

ðπ t2 ez cos t cosðν tÞ dt 0 ∞ ð

(3:4:31) e

0

−z cosh t −ν t

½ðπ − t Þ sinðπ νÞ + 2 π t cosðπ νÞ dt 2

2

3.4 Higher Order Derivatives of the Bessel and Related Functions

127

ðπ

∂ 3 Iν ðzÞ 1 = ∂ν3 π

t3 ez cos t sinðν tÞ dt 0

1 − π

∞ ð

(3:4:32) e−z cosh t −ν t ½ðt3 − 3 π2 tÞ sinðν tÞ + ðπ3 − 3π t2 Þ cosðν tÞ dt

0

From the integral representations of Iν(z) given in (3.3.37) and that of Kν(z) in (3.3.38) ∞ ð

Kν ðzÞ =

e−z cosh t coshð νtÞ dt

;

Re z > 0

(3:4:33)

0

we have the expected result by using (3.4.30) ∞   ð ∂ Iν ðzÞ = − e−z cosh t dt = − K0 ðzÞ ∂ν ν = 0

(3:4:34)

0

Differentiation of (3.4.33) with respect to ν gives ∂ n Kν ðzÞ = ∂νn ∂ Kν ðzÞ = ∂νn n

∞ ð

0 ∞ ð

tn e−z cosh t coshðνtÞ dt

; n = 0, 2, 4, 6, . . .

(3:4:35)

tn e−z cosh t sinhðνtÞ dt

; n = 1, 3, 5, 7, . . .

(3:4:36)

0

and ∞  n  ð ∂ Kν ðzÞ = tn e−z cosh t dt ∂νn ν =0

; n = 0, 2, 4, 6, . . .

(3:4:37)

0

 n  ∂ Kν ðzÞ =0 ∂νn ν =0

;

n = 1, 3, 5, 7 . . .

(3:4:38)

Similar integrals as in (3.4.33), define particular cases of modified Lommel functions ∞ ð

S0, ν ðzÞ =

e−z sinh t coshðνtÞ dt

;

Re z > 0

(3:4:39)

0

and therefore we have ∂ n S0, ν ðzÞ = ∂νn

∞ ð

0

tn e−z sinh t coshðνtÞ dt

;

n = 0, 2, 4, 6, . . .

(3:4:40)

128

3 Differentiation and Integration with Respect to the Order of the Bessel

∞ ð

∂ n S0, ν ðzÞ = ∂νn

tn e−z sinh t sinhðν tÞ dt

;

n = 1, 3, 5, 7, . . .

(3:4:41)

0

with ∞  n  ð ∂ S0, ν ðzÞ = tn e−z sinh t dt ∂νn ν =0

;

n = 0, 2, 4, 6, . . .

(3:4:42)

0

 n  ∂ S0, ν ðzÞ =0 ; ∂νn ν =0

n = 1, 3, 5, 7 . . .

(3:4:43)

Since ∞ ð

S1, ν ðzÞ = z

e−z sinh t coshðνtÞ cosh t dt

;

Re z > 0

(3:4:44)

0

similar expressions, as (3.4.40)–(3.4.33), can be written for the function S1,ν(z). In case of the Lommel functions which have the same orders we have ∞ ð

Sν, ν ðzÞ =

e

−z sinh t

∞ ð

2 ν

½z ðcosh tÞ  dt =

0

e−z sinh t + ν ½ln z + 2 lnðcosh tÞ dt (3:4:45)

0

Re z > 0 and higher derivatives with respect to the order are ∂ n Sν, ν ðzÞ = zν ∂νn

∞ ð

e−z sinh t ðcosh tÞ2 ν ½ln z + 2 lnðcosh tÞn dt

0

(3:4:46)

n = 1, 2, 3, . . . For ν = 0 it becomes ∞  n  ð ∂ Sν, ν ðzÞ = e−z sinh t ½ln z + 2 lnðcosh tÞn dt ∂νn ν =0

(3:4:47)

0

and in this case the first derivative with respect to the order is ∞ ∞   ð ð ∂ Sν, ν ðzÞ = ln z e−z sinh t dt + 2 e −z sinh t lnðcosh tÞ dt ∂ν ν =0 0

∞ ð

= ln z S0, 0 ðzÞ + 2 0

0

pffiffiffiffiffiffiffiffiffiffiffi e pffiffiffiffiffiffiffiffiffiffiffi lnð 1 + x2 Þ dx 1 + x2 −z x

(3:4:48)

129

3.4 Higher Order Derivatives of the Bessel and Related Functions

The integral representation of the Lommel function with the same orders sν,ν(z) is pffiffiffi sν, ν ðzÞ = π zν

π=2 ð

ðcos tÞ2 ν sinðz sin tÞ dt

; Re ν > −

1 2

(3:4:49)

0

and therefore ∂ n sν, ν ðzÞ = zν ∂νn

π=2 ð

ðcos tÞ2 ν ½ln z + 2 lnðcos tÞn sinðz sin tÞ dt

0

(3:4:50)

n = 1, 2, 3, . . . with π=2  n  ð ∂ sν, ν ðzÞ = ½ln z + 2 lnðcos tÞn sinðz sin tÞ dt ∂νn ν =0

(3:4:51)

0

For ν = 0, the first derivative with respect to the order is π=2 π=2   ð ð ∂ sν, ν ðzÞ = ln z sinðz sin tÞ dt + 2 lnðcos tÞ sinðz sin tÞ dt ∂ν ν =0 0

0

ð1 = ln z s0, 0 ðzÞ + 2 0

pffiffiffiffiffiffiffiffiffiffiffi lnð 1 − x2 pffiffiffiffiffiffiffiffiffiffiffi sinðz xÞ dx ð 1 − x2 Þ

(3:4:52)

Since the Lommel functions with the same orders are interrelated with the Struve functions   pffiffiffi 1 ν−1 H ν ðzÞ (3:4:53) sν, ν ðzÞ = 2 πΓ ν+ 2   pffiffiffi 1 ν−1 ½H ν ðzÞ − Yν ðzÞ (3:4:54) Sν, ν ðzÞ = 2 πΓ ν+ 2 it is possible to derive higher derivatives with respect to the order of the Struve functions. However, differentiation starts to be arduous with increasing value of n, and (3.4.53) and (3.4.54) are actually convenient only for the first derivatives of the Struve functions. The higher order derivatives with respect to the order of the Anger and Weber functions can directly be determined by taking into account their definitions ðπ 1 cosðν t −z sin tÞdt J ν ðzÞ = π 0

(3:4:55)

130

3 Differentiation and Integration with Respect to the Order of the Bessel

and therefore, ðπ ∂ n J ν ðzÞ ð− 1Þn=2 n = t cosðνt −z sin tÞ dt π ∂νn

;

0, 2, 4, 6, . . .

(3:4:56)

;

1, 3, 5, 7, . . .

(3:4:57)

0, 2, 4, 6, . . .

(3:4:58)

0

ðn + 1Þ=2

∂ n J ν ðzÞ ð− 1Þ = π ∂νn

ðπ tn sinðνt −z sin tÞ dt 0

For ν = 0 we have  n  ðπ ∂ J ν ðzÞ ð− 1Þn=2 n = t cosðz sin tÞdt π ∂νn ν =0

;

0

 n  ðπ ∂ J ν ðzÞ ð− 1Þðn + 1Þ=2 n =− t sinðz sin tÞ dt π ∂νn ν =0

;

1, 3, 5, 7, . . .

(3:4:59)

0

Using the integral representation of the Weber function Eν ðzÞ =

ðπ 1 sinðνt −z sin tÞ dt π

(3:4:60)

0

Higher derivatives with respect to the order are ðπ ∂ n Eν ðzÞ ð− 1Þn=2 n = t sinðν t −z sin tÞdt π ∂νn

;

0, 2, 4, 6, . . .

(3:4:61)

0

ðn + 1Þ=2

∂ n Eν ðzÞ ð− 1Þ = π ∂νn

ðπ tn cosðν t −z sin tÞ dt

; 1, 3, 5, 7, . . .

(3:4:62)

0, 2, 4, 6, . . .

(3:4:63)

0

and for ν = 0 we have  n  ðπ ∂ Eν ðzÞ ð− 1Þn=2 n = t sinðz sin tÞdt π ∂νn ν =0

;

0

 n  ðπ ∂ Eν ðzÞ ð− 1Þðn + 1Þ=2 n = − t cosðz sin tÞ dt π ∂νn ν =0

; 1, 3, 5, 7, . . .

(3:4:64)

0

As expected, the first derivatives with respect to the order determined above by using the Schlaefli integral representations of the Anger and Weber functions are identical to those expressions derived in (3.4.61)–(3.4.64).

3.4 Higher Order Derivatives of the Bessel and Related Functions

131

Finally, it is possible to obtain the higher order derivatives with respect to the order of the integral Bessel functions Jiν(z) and Kiν(z). From the integral representation [4] 1 1 Jiν ðzÞ = + 2ν π ν

ðπ cot t sinðz sin t −ν tÞ dt 0

sinðπ νÞ − πν

∞ ð

e−z sinh t −νt coth t dt

(3:4:65)

0

Re ν > 0 by direct differentiation of (3.4.65) we have ∂ n Jiν ðzÞ ð− 1Þn n! 1 = + n n + 1 ∂ν 2ν πν

ðπ tn

ð− 1Þn=2 ½1 + ð− 1Þn  sinðz sin t −ν tÞ 2

0

 ð− 1Þðn − 1Þ=2 ½1 − ð− 1Þn  + cosðz sin t −ν tÞ dt 2 ∞  ð Fn ðνÞ sinðπ νÞ tn −z sinh t −νt e coth tdt − + ð− 1Þn πν π 0

n = 0, 1, 2, 3, . . .

(3:4:66)

where Fn(ν) are products of two functions f(ν) and g(ν) defined in (3.4.67). It follows from the Leibniz theorem for differentiation of a products that f ðνÞ =

1 ν

;

gðνÞ = sinðπ νÞ

! n n dn − k f ðνÞ d k gðνÞ dn ½f ðνÞgðνÞ X = Fn ðνÞ = dνn d νn − k dνk k k =0 ! n X n = an − k ðνÞ bk ðνÞ k k =0   ðn − kÞ! πk an − k ðνÞ = ð− 1Þn − k n − k + 1 ; bk ðνÞ = πk sin π ν + 2 ν

(3:4:67)

Glasser suggested to express Fn(ν) in an alternative way, in terms of elementary trigonometric integrals

132

3 Differentiation and Integration with Respect to the Order of the Bessel

dn Fn ðνÞ = n dν



 ð1 sinðπ νÞ πn + 1 = tn cosðπ ν tÞ dt 2 ν −1

(3:4:68)

n = 0, 2, 4, 6, ... and dn Fn ðνÞ = n dν

  π n ð1 sinðπ νÞ πn + 1 = tn sinðπ ν tÞ dt sin 2 ν 2 −1

n = 0, 1, 3, 5, ... The first three Fn(ν) functions are F1 ðνÞ =

π cosðπ νÞ sinðπ νÞ − ν ν2

F2 ðνÞ = −

π2 sinðπ νÞ 2π cosðπ νÞ 2 sinðπ νÞ − − ν ν2 ν3

F3 ðνÞ = −

π3 cosðπ νÞ 3π2 sinðπ νÞ 6 π cosðπ νÞ 6 sinðπ νÞ + − + ν ν2 ν3 ν4

(3:4:69)

Since the integral representation of the modified Bessel function of the second kind has a simple form ∞ ð

Kν ðzÞ =

e−z cosh t coshðνtÞdt

(3:4:70)

0

and the integral function Kiν(z) is defined by ∞ ð

Kiν ðzÞ =

Kν ðxÞ dx x

(3:4:71)

z

it is possible to derive higher derivatives with respect to the order by introducing (3.4.70) into (3.4.71) # ∞ ð "∞ ð 1 −x cosh t e coshðν tÞ dt dx (3:4:72) Kiν ðzÞ = x z

0

Changing order of integration in (3.4.72) we have "∞ ∞ ð ð −x cosh t # e Kiν ðzÞ = coshðνtÞ dx dt x 0

z

(3:4:73)

3.4 Higher Order Derivatives of the Bessel and Related Functions

133

The inner integral in (3.4.73) is the exponential integral [9] ∞ ð

E1 ða zÞ =

e− a t dt = − Eið− azÞ t

(3:4:74)

z

and therefore the integral representation of Kiν(z) is given by ∞ ð

Kiν ðzÞ =

E1 ðz cosh tÞ coshðν tÞ dt

(3:4:75)

0

Direct differentiation of (3.4.75) with respect to the order gives the higher derivatives ∂n Kiν ðzÞ = ∂ νn

∞ ð

tn E1 ðz cosh tÞ coshðνtÞ dt (3:4:76)

0

n = 0, 2, 4, 6, ... and ∂ n Kiν ðzÞ = ∂ νn

∞ ð

tn E1 ðz cosh tÞ sinhðνtÞ dt (3:4:77)

0

n = 1, 3, 5, 7, ... For ν = 0, we have analogous expressions to those in (3.4.37) and (3.4.38) ∞  n  ð ∂ Kiν ðzÞ = tn E1 ðz cosh tÞ dt ; ∂νn ν =0 0  n  ∂ Kiν ðzÞ = 0 ; n = 1, 3, 5, 7, ... ∂νn ν =0

n = 0, 2, 4, 6, . . . (3:4:78)

4 Mathematical Operations with Respect to the Order of the Bessel and Related Functions – Integration, Differentiation, Series and Limits The Application of Laplace Transform Methods and Other Related Topics 4.1 Integration with Respect to the Order of the Bessel and Related Functions Contrary to derivatives with respect to the order which were sporadically investigated in mathematical literature, at least with an enduring interest, the integration with respect to the order appeared only on very special occasions. Usually, this occurred as a by-product of solving of mathematical problems or illustrating some mathematical techniques. Only a few such cases can be found, and they all will be presented here. Besides, the next section (Section 4.2) will discuss the systematic application of the Laplace transformation to the integration with respect to the order. As pointed out above, a number of known integrals with respect to the order of Bessel functions is small, but exists as a large group of integrals when the order is an imaginary number, they are connected with the Kontorovich-Lebedev or with other integral transforms [18]. These integrals are entirely omitted here. The first example of integration with respect to the order is quoted by Watson in his book [7] He considered one of the extraordinary integrals derived by Ramanujan in 1920 ∞ ð

IðtÞ = −∞

Jμ + ξ ðxÞ Jν − ξ ðxÞ i t ξ e dξ xμ + ξ xν − ξ

x, y > 0 ;

(4:1:1)

Re ðμ + νÞ > 1

and for real values of t, this integral is

(

IðtÞ =

2 cosðt=2Þ

ðμ + νÞ=2

x2 e − i t=2 + y2 ei t=2

ei t ðν − μÞ=2

0 : jtj < π sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   t ϕ = 2 cos ½x2 e− i t=2 + y2 ei t=2  2 For t = 0 and y = x it reduces to

https://doi.org/10.1515/9783110681642-004

Jμ + ν ðϕÞ

;

jtj < π (4:1:2)

4.1 Integration with Respect to the Order of the Bessel and Related Functions

135

∞ ð

Ið0Þ =

J μ + ξ ðxÞ J ν − ξ ðxÞ d ξ = J μ + ν ð2 xÞ

(4:1:3)

−∞

In 1954, Cooke [46] considered only cosine function in (4.1.1) and derived it in a way which was different from the Ramanujan result by starting from the integral which represents the product of two Bessel functions [7] ∞ ð

Jμ ðzÞ Jν ðzÞ =

Jμ + ν ð2 z cos θÞ cos½ðμ − νÞθ dθ

(4:1:4)

0

Reðμ + νÞ > − 1 He replaced the independent variable and orders of the Bessel functions in the following way t = 2θ, μ = α + ξ and ν = β – ξ. In the next step, Cooke multiplied both sides of (4.1.4) by cos(фξ) and integrated from − ∞ to + ∞ with respect to ξ. The result of integration is ∞ ð

Jα + ξ ðzÞ Jβ − ξ ðzÞ cosðϕ ξÞ d ξ −∞

=

8 h i h  i < cos α − β ϕ Jα + β 2z cos ϕ ; z 2 :

0

(4:1:5)

jϕj < π

; jϕj > π

where for ф = 0 and β = α (4.1.5) is identical with (4.1.2). Using the same procedure Cooke derived few more integrals with respect to the order for the modified Bessel and the Anger functions, and they are presented below: ∞ ð

Iν ðzÞ cosðθνÞ dν =

IðθÞ = 0

1 − 2π

(

∞ ð

e

∞ ð

Ið0Þ =

ðπ + θÞ2 + t2

1 2πi

;

jarg zj ≤

ez Iν ðzÞ dν = − 2

0 α +ði ∞

α−i∞

π+θ

− z cosh t

0

jReνj > 0

ez cos θ 2

∞ ð

0

1 2

+

π−θ

)

ðπ − θÞ2 + t2

dt

(4:1:6)

; jθj < π

e− z cosh t dt π 2 + t2

(4:1:7)

Kν ðzÞ cosh½ðν − αÞθ dν = e − z cosh θ coshðαθÞ (4:1:8) π jarg zj < 2

136

4 Mathematical Operations with Respect to the Order of the Bessel

1 2πi

1 2πi

α +ði ∞

Kν ðzÞ sinh½ðν − αÞθ dv = − e − z cosh θ sinhðαθÞ

(4:1:9)

α−i∞ α +ði ∞

Kν ðzÞ dv = e − z

(4:1:10)

α−i∞

8 > < cosðz sin θÞ ; j θ j < π Jν ðzÞ cosðθνÞ dν = 1=z ; j θ j = π > : −∞ 0 ; jθj > π ∞ ð

(

∞ ð

Jν ðzÞ sinðθνÞ dν =

sinðz sin θÞ 0

−∞

;

; jθj < π jθj > π

(4:1:11)

(4:1:12)

and ∞ ð

Jν ðzÞ dν = 1

(4:1:13)

−∞

where the last integral is valid for any value of argument z. The Cooke results given in (4.1.6) were expressed in terms of an infinite series by Fényes [50, 51] in 1993. He stated that such integrals are important in the heatconduction problems, when solutions of an integral equation of convolution type are required. ∞ ð

Iν ðzÞ sinðθνÞ dν = 0

  ez cos θ − e − θ π + θ ln 2π π − θ " # ∞ ð z t2 + ðθ − πÞ2 − z cosh t e ln sinh t dt − 4π t2 + ðθ + πÞ2 0

ðz SiðzÞ = 0

and

sin t dt t

∞ 1 X Ik ðzÞ cosðkθÞfc ½ðπ − θÞk − c½ðπ + θÞ k g + π k=1 ∞ 1 X + Ik ðzÞ sinðkθÞfSi ½ðπ − θÞk + Si½ðπ + θÞ k g π k=1 ðz 1 − cos t ; cðzÞ = dt t 0

(4:1:14)

4.1 Integration with Respect to the Order of the Bessel and Related Functions

∞ ð

ν Iν ðzÞ dν = 0

e− z −2 π2 −z

e = 2 −2 π

∞ ð

0 ∞ ð

0

t e − z cosh t ðπ2 + t2 Þ2

+ dt +

t e − z cosh t

t + 2 2 2 π ðπ + t Þ

ðπ

∞ 2 X k SiðπkÞIk ðzÞ π k=1

137

(4:1:15)

dt ez cos t sin t t

0

Considering heat transfer in cylindrical coordinates, Fényes solved the following integral equation ðτ f ðtÞ K0 ðτ − tÞ dt = 1

;

τ>0

(4:1:16)

0

and found that its solution contains the convolution operation f ðτÞ ¼

1 τ

∞ ð

ν Iν ðτÞ dν  0

1 τ

ðτ

∞ ð

ν Iν ðτÞ dν  0

I1 ðxÞ dx x

(4:1:17)

0

Introducing (4.1.15) into (4.1.17), the solution of (4.1.16) can be written as f ðτÞ =

∞ 2 X k Siðπ kÞ Ik ðτÞ πτ k = 1

ðτ ∞ 2 X Ik + 1 ðtÞ ðk + 1Þ Siðπ kÞ Ik ðτÞ − dt π k=1 t e− τ 2 + 2 − πτ τ

∞ ð

0

(4:1:18)

0

( ) ðτ ∞ ð e− τ 2 t e − τ cosh t I1 ðtÞ dt − dt  − dt π2 τ τ ðπ2 + t2 Þ2 t ðπ2 + t2 Þ2

t e− τ cosh t

0

0

If the modified Bessel function of the first kind Iν(z) is replaced by the Bessel functions Jν(z), Fényes found that ∞ ð

Jν ðzÞ cosðθνÞ dν = 0

  cosðsin θÞ sinðsin θÞ π − θ + ln 2 2π π + θ ( ) ∞ ð 1 π+θ π−θ − z sinh t e + dt − 2π t2 + ðθ + πÞ2 t2 + ðπ − θÞ2 0

∞ 1 X J2 k + 1 ðzÞ sin½ð2 k + 1Þθ Ak ðθÞ + π k=0 ∞ 1 X + J2 k + 1 ðzÞ cos½ð2 k + 1Þ θ Bk ðθÞ π k=0

(4:1:19)

138

4 Mathematical Operations with Respect to the Order of the Bessel

where Ak ðθÞ = c ½ðπ + θÞð2 k + 1Þ − c ½ðπ − θÞð2 k + 1Þ

(4:1:20)

Bk ðθÞ = Si½ðπ + θÞð2 k + 1Þ + Si½ðπ − θÞ ð2k + 1Þ

The expression (4.1.19) is valid for │θ│< π and z ≥ 0, for │θ│> π, the first term is replaced by zero, and for θ = π, the value of first term is 1/4 and the second term is zero. If θ = 0, the integral in (4.1.19) becomes ∞ ð

1 Jν ðzÞ dν = − 2

∞ ð

0

0

∞ e − z sinht 2X dt + Si½ð2k + 1Þπ J2k + 1 ðzÞ π2 + t2 π k=0

(4:1:21)

and similar integral in (4.1.15) is ∞ ð

ν Jν ðzÞ dν = 0

1 z + −2 π2 2

∞ ð

0

t e − z sinh t ðπ2 + t2 Þ2

dt +

∞ 4X k Sið2π kÞJ2 k ðzÞ π k=1

(4:1:22)

Fényes [52, 76] also derived integral representations of moments of the Bessel functions which he defined by ∞ ð

Mn ðtÞ =

ν n Jν ðtÞ dt

;

t>0 ;

n = 0, 1, 2, 3, . . .

(4:1:23)

ν n Iν ðtÞ dt

;

t>0 ;

n = 0, 1, 2, 3, . . .

(4:1:24)

0 ∞ ð

mn ðtÞ = 0

and the Laplace transforms of these moments by 1 JνðnÞ ðt, sÞ = π

"∞ ð

ðπ cosðt sin xÞ 0

1 + π 1 − π

# e − ν s νn cosðνxÞ dν dx

0

"∞ ð

ðπ sinðt sin xÞ 0 ∞ ð

0

"∞ e − t sinh x

ð

0

# e − ν s νn sinðνxÞ dν dx

(4:1:25)

#

e − ν s sinðνxÞ dν dx

0

These Laplace transforms are based on the Schlaefli integral representations of the Bessel functions. Performing the change of order of integration in (4.1.25) we have

4.1 Integration with Respect to the Order of the Bessel and Related Functions

IνðnÞ ðt, sÞ =

1 π −

"∞ ð

ðπ et cos x 0

1 π

# e − ν s νn cosðνxÞ dν dx

0 ∞ ð

139

"∞

e − t cosh x

0

ð

#

(4:1:26)

e − ν ðx + sÞ νn dν dx

0

where the inner integrals in (4.1.25) and (4.1.26) are known [36] ∞ ð

e − ν s νn sinðνxÞ dν = ð− 1Þn

0

∂n  x ∂ sn s2 + x 2



s n+1 X ð− 1Þk = n! 2 2 s +x 0 ≤ 2k ≤ n ∞ ð

e − ν s νn cosðνxÞ dν = ð− 1Þn

0

!  x 2k + 1 s 2k + 1

(4:1:27)

! n + 1 x 2k s 2k

(4:1:28)

n+1

∂n  s ∂ sn s 2 + x 2



X s n + 1 ð− 1Þk = n! 2 2 s +x 0 ≤ 2k ≤ n + 1 ∞ ð

e − ν ðs + 1Þ νn dν =

0

n!

(4:1:29)

ðs + xÞn + 1

In such way, Fényes derived general expressions for the Laplace transforms of the moments in (4.1.23) and (4.1.24). However, his interest was limited to behaviour of the moments for large times, t ! ∞, not to the inversion of the Laplace transforms. This is equivalent for behaviour of the Laplace transforms for s ! 0 ∞ ð

Mn ðtÞ =

νn Jν ðtÞ dν = Qn ðtÞ 0

2n + 2 n! + pffiffiffiffiffiffiffiffi 2π t πn + 1

   πn π + oð1Þ sin t − − 2 4

(4:1:30) ; t!∞

Using the polynomials Qn(t), Fényes derived the first 11 moments in an explicit form

140

4 Mathematical Operations with Respect to the Order of the Bessel

Q0 ðtÞ = 1

;

Q3 ðtÞ = t3 + t

Q1 ðtÞ = t ;

;

Q2 ðtÞ = t2

Q4 ðtÞ = t4 + 4t2

Q6 ðtÞ = t6 + 20 t4 + 16t2

;

;

Q5 ðtÞ = t5 + 10t3 + t

Q7 ðtÞ = t7 + 35t5 + 191t3 + t

Q8 ðtÞ = t8 + 56 t6 + 336t4 + 64t2

(4:1:31)

Q9 ðtÞ = t9 + 84t7 + 996 t5 + 820t3 + t Q10 ðtÞ = t10 + 120t8 + 2352t6 + 544t4 + 256t2 In case of moments of the modified Bessel function of the first kind, the final results of his calculations are   et et t1=2 1 ; t!∞ m0 ðtÞ ⁓ ; m1 ðtÞ⁓ pffiffiffiffiffiffi 1 − 2 24t 2π   et t et t3=2 1 7 m2 ðtÞ ⁓ ; m3 ðtÞ⁓ pffiffiffiffiffiffi 2 + + 2 4t 960 t2 2π   et ðt + 3t2 Þ et t5=2 5 16 31 m4 ðtÞ ⁓ ; m5 ðtÞ⁓ pffiffiffiffiffiffi 8 + + 2 − 2 t t 8064t3 2π (4:1:32) et ðt + 15 t2 + 15 t3 Þ m6 ðtÞ ⁓ 2   t 7=2 et 50 91 64 127 m7 ðtÞ ⁓ pffiffiffiffiffiffi 48 + + 2+ 3 + t 8t t 30720t4 2π m8 ðtÞ ⁓

et ðt + 63t2 + 210 t3 + 105 t4 Þ 2

A number of integrals of the modified Bessel functions of the second kind and the imaginary order is presented in [18], the simplest two are ∞ ð

Kα − i ν ðaÞKi ν − β ðbÞ dν = π Kα + β ða + bÞ −∞

(4:1:33)

Re a, b > 0 and ∞ ð

K i ν ðaÞKi ν ðbÞ cosh½ðπ − jθjÞ νdν =

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π K0 ð a2 + b2 − 2 ab cos θÞ 2

0

0 ≤ θ ≤ 2π Recently, Becker [88] in 2009 demonstrated that

(4:1:34)

4.1 Integration with Respect to the Order of the Bessel and Related Functions

∞ ð

−∞ ∞ ð

−∞ ∞ ð

141

Ji ν ði zÞ eπ ν=2 dν = − i e − z sinhðπ νÞ

(4:1:35)

Ii ν ðzÞ dν = − ie − z sinhðπ νÞ

(4:1:36)

Ki ν

z π dν = e − z=2 2 2

(4:1:37)

0 ∞ ð

ν Γðλ + i νÞ Γðλ + iνÞ Ii ν ðzÞ dν −∞

=

" # 8 λ+1 pffiffiffi  >

:

pffiffiffi  − 2λ π i Γ λ + 21 zλ e − z

;

;

− 1 < Re λ ≤ 0

(4:1:38)

Re λ > 0

∞ ð

ν sinhðπ νÞΓðλ + i νÞ Γðλ + iνÞ Ki ν ðzÞ dν 0

=

" # 8  λ+1 > < − 2λ − 1 π3=2 Γ λ + 21 Γð2 − λÞ Kλ ðzÞ − zλ e − z ;

> :

 2λ − 1 π3=2 Γ λ + 21 zλ e − z

;

− 1 < Re λ ≤ 0

(4:1:39)

Re λ > 0

These integrals have some application in a time-dependent radiation transport theory. Finally, it is worthwhile to mention an equality of integrals proved by van der Pol [82]. ∞ ð

−∞

K ν ðaÞ e − b cosh ν dν =

∞ ð

K ν ðbÞ e − a cosh ν dν

−∞

(4:1:40)

Re a, b > 0 All integrals presented in Section 4.1 were evaluated in a rather long and messy way, and this shows that the operation of the integration is considerably more complex than differentiation with respect to the order of Bessel functions.

142

4 Mathematical Operations with Respect to the Order of the Bessel

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function of the First Kind by Applying the Laplace Transform Approach The use of the convolution (product) theorem of the Laplace transformation to determine derivatives with respect to the order, in a compact and elegant form, was initiated by van der Pol [81] in 1929. The finding of inverse transforms in the Laplace transformation is facilitated if in the Laplace transformation ∞ ð

Lff ðtÞg =

e − s t f ðtÞ dt = FðsÞ

(4:2:1)

0

1 f ðtÞ = L fFðsÞg = 2πi −1

c +ði ∞

e s t FðsÞ ds c−i∞

(4:2:2)

Re s > c the transforms F(s) can be written as a product F(s) = F1(s)·F2(s) of two functions of s and inverses L−1{F1(s)} = f1(t) and L−1{F2(s)} = f2(t) are known. Then, the inverse f(t), the original function, is given by the convolution integral f ðtÞ = L − 1 fFðsÞg = f1 ðtÞf2 ðtÞ ðt

ðt f1 ðuÞf2 ðt − uÞdu =

= 0

f1 ðt − uÞf2 ðuÞdu

(4:2:3)

0

t>0 where the convolution of two functions f1(t) and f2(t) is denoted by symbol. The convolution of two or more functions obeys the commutative law and the associative law and is distributive with respect to the addition [31]. Van der Pol [82] started with the Laplace transforms of the Bessel function of the first kind of the order ν and the integral Bessel function of the order zero 1 LfJν ðtÞg = pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν 2 s + 1 ðs + s2 + 1Þ ; Re ν > − 1 pffiffiffiffiffiffiffiffiffiffiffi ln ðs + s2 + 1Þ LfJi0 ðtÞg = − s

(4:2:4)

Re s > 0

Re s > 0 and he differentiated the Laplace transform in (4.2.4) with respect to ν

(4:2:5)

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function

 pffiffiffiffiffiffiffiffiffiffiffi

 ln s + s2 + 1 ∂ Jν ðtÞ = − pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffiffiffiffiffiffiffiffi!ν ∂ν s2 + 1 s + s2 + 1 Re s > 0

;

143

(4:2:6)

Re ν > − 1

If both Laplace transforms from (4.2.5) and (4.2.6) are written as a product in the following way h pffiffiffiffiffiffiffiffiffiffiffii

 ln s + s2 + 1 ∂ Jν ðtÞ s =− (4:2:7) L · pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi!ν s ∂ν 2 s + 1 s + s2 + 1 the inverse according to (4.2.3) is ∂ Jν ðtÞ = ∂ν

ðt

Ji0 ðt − xÞ J ′ν ðxÞ dx

(4:2:8)

0

Re ν > 0 where the derivative of the Bessel function with respect to argument t is ðt ∂ Jν ðtÞ 1 Ji0 ðt − xÞ ½Jν − 1 ðxÞ − Jν + 1 ðxÞ dx = ∂ν 2 0

(4:2:9)

Re ν > 0 It comes from the operator s in the Laplace transform domain and is inversed by the Duhamel integral with Jν(0) = 0 for Re ν > 0. For the integral Bessel function see paper of Humbert [89]. By arranging the Laplace transforms in different way, van der Pol obtained alternative expressions for the first derivative with respect to the order of the Bessel function of the first kind      ðt  ν ∂ Jν ðtÞ 1 t x − ln Jν ðtÞ + 2 Ciðt − xÞJν − 1 ðxÞ dx = ψ ν− ∂ν 2 2 t ∞ ð

CiðzÞ = −

0

cos t dt t

(4:2:10)

; Re ν > 0

z

    ðt  ν  ∂ Jν ðzÞ t x x2 Jν ðtÞ + ln 1 − 2 Jν − 1 ðxÞ dx = γ + ln t ∂ν 2 t 0

Re ν > 0

(4:2:11)

144

4 Mathematical Operations with Respect to the Order of the Bessel

and ∂ Jν ðtÞ 1 = PV ∂ν 2

∞ ð

−∞

eνjxj Jν ðt e − x=2 Þ dx ex − 1

(4:2:12)

Re ν > − 1 After a rather long period of time, only in 1985, the present author and the late Naftali Kravitsky [4] used the Laplace transformation, but this time in a more systematic way to obtain integral representations of derivatives and integrals with respect to the order of the Bessel functions, the integral Bessel function and the Anger function. This was performed by an evaluation of the complex integrals given in (4.2.2) by using different modifications of the Bromwich contour and by applying various operational rules of the Laplace transformation. These operational methods were later extended to the Struve and Kelvin functions during the period 1989–1991 [5, 6]. Results from these investigations and applied mathematical procedures are discussed below. Let us start with determination of the following integral with respect to the order ν ∞ ð

Jλ ðtÞ dλ

Iðν, tÞ =

(4:2:13)

ν

The Laplace transform of the Bessel function of the first kind is 1 1 LfJν ðtÞg = pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν = pffiffiffiffiffiffiffiffiffiffiffi 2 s + 1 Qν s2 + 1 ðs + s2 + 1Þ pffiffiffiffiffiffiffiffiffiffiffi Q = ðs + s2 + 1Þ ; Re s > 0 ; Re ν > − 1

(4:2:14)

and by performing integration under the integral sign, we have (∞ ) ð LfIðν, tÞg = L

Jλ ðtÞ dλ ν

1 = pffiffiffiffiffiffiffiffiffiffiffi 2 s +1

∞ ð

ν

(4:2:15) dλ 1 ffi pffiffiffiffiffiffiffiffiffiffiffi λ = pffiffiffiffiffiffiffiffiffiffi 2 2 s + 1 Qν ln Q ðs + s + 1Þ

The inverse of (4.2.15) is given by the complex integral (4.2.2) ∞ ð

Jλ ðtÞ dλ =

Iðν, tÞ = ν

1 2πi

c +ði ∞

c−i∞

e st pffiffiffiffiffiffiffiffiffiffiffi ds s2 + 1 Qν ln Q

(4:2:16)

Re s > c Usually, such complex integrals which have poles, branch points and essential singularities are evaluated with the help of closed, different Bromwich contours, the Cauchy residue theorem and the Jordan lemma. Suitable for complex integration of (4.2.16), the Bromwich contour is presented in Figure 4.1.

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function

145

B

s-plane

R→∞ +i E

F G

D

C L

c

O

K

H

J

I

–i

A

Figure 4.1: The Bromwich contour used in complex integration of equation (4.2.16).

The integrand in (4.2.16) has two conjugate branch points at s = ± i and a pole at the origin. Contributions to the integral come from CD, DE, EG, HI, JK and KL parts of the contour and a half residue at the pole s = 0. The contributions from BC and LA circular arcs with infinite radius R in an imaginary plane vanish according to the Jordan lemma. The sum of these contributions is ∞ ð

Jλ ðtÞ dλ =

Iðν, tÞ = ν

1 − π

∞ ð

e

1 1 + 2 π

ðπ sinðt sin x − ν xÞ 0

− t sinh x − ν x

0

dx x (4:1:17)

½π cosðπ νÞ + x sinðπ νÞ dx π 2 + x2

For an entire interval, starting from ν = 0, the integral in (4.2.17) reduces to ∞ ð

Ið0, tÞ = 0

1 1 Jλ ðtÞ dλ = + 2 π

and it also follows that

ðπ 0

dx sinðt sin xÞ − x

∞ ð

0

e − t sinh x dx π 2 + x2

(4:2:18)

146

4 Mathematical Operations with Respect to the Order of the Bessel

ðν Jλ ðtÞ dλ = Ið0, tÞ − Iðν, tÞ

(4:2:19)

0

The rules of the Laplace transformation permit easily to obtain the limiting behaviour of the integral (4.2.18) for small or large values of t. From (4.2.15) we have (∞ ð

)

1 Jλ ðtÞ dλ = pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi! 2 s + 1 ln s + s2 + 1

LfIð0, tÞg = L 0

(4:2:20)

and therefore, LfIð0, t ! ∞Þg⁓

1 1 ⁓ lnð1 + sÞ s

;

s!0

(4:2:21)

which means

 1 =1 s

;

t!∞

(4:2:22)

1 s lnð2sÞ

;

s!∞

(4:2:23)

Ið0, t ! ∞Þ⁓L − 1 Similarly LfIð0, t ! 0Þg⁓

The inverse of (4.2.23) can be expressed in terms of the Volterra function [30] Ið0, t ! 0Þ⁓L− 1

   1 t =ν s lnð2sÞ 2

;

t!0

 λ   ∞ t ð 2 t ν = dλ Γðλ + 1Þ 2

(4:2:24)

0

and its asymptotic expression for small t is known [4] 1 Ið0, t ! 0Þ⁓ lnð2=tÞ

(

ψð1Þ ½ψð1Þ2 − ψ′ð1Þ − ... 1− + lnð2=tÞ ½lnð2=tÞ2

) (4:2:25)

t!0 As expected, a direct differentiation of (4.2.17) leads to the Schlaefli representation of the Bessel function of the first kind

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function

∂ Jν ðtÞ = − ∂ν

(∞ ð

) Jλ ðtÞ dλ =

ν

sinðπ νÞ − π

∞ ð

1 π

147

ðπ cosðt sin x − ν xÞ dx 0

(4:2:26)

e − t sinh x − ν x dx

0

Returning to the Laplace transform of derivative given in (4.2.6)  pffiffiffiffiffiffiffiffiffiffiffi

 ln s + s2 + 1 ∂ Jν ðtÞ L = − pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi!ν ∂ν s2 + 1 s + s2 + 1 Re s > 0

;

(4:2:27)

Re ν > − 1

its inverse is known for ν = 0 [36]  pffiffiffiffiffiffiffiffiffiffiffi 9 8   < ln s + s2 + 1 = π ∂ Jν ðtÞ pffiffiffiffiffiffiffiffiffiffiffi = L−1 − = Y ðtÞ : ; 2 0 ∂ν ν = 0 s2 + 1

(4:2:28)

and the expected result is reached. The Laplace transforms of higher derivatives with respect to the order can be determined by direct differentiation of (4.2.27), but the operational substitution rule gives a more elegant way to obtain them. If a function f(t) has the Laplace transform F(s) then [31] (∞ ) ð h  pffiffiffiffiffiffiffiffiffiffiffi i 1 1 f ðλÞ Jλ ðtÞ dλ = pffiffiffiffiffiffiffiffiffiffiffi F ln s + s2 + 1 = pffiffiffiffiffiffiffiffiffiffiffi Fðln QÞ (4:2:29) L 2 2 s +1 s +1 0

If in this substitution formula the chosen function is for example f(t) = 1, then F(s) = 1/s, and from (4.2.29) it follows that (∞ ) ð 1 1 Jλ ðtÞ dλ = pffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffi = pffiffiffiffiffiffiffiffiffiffiffi (4:2:30) L 2 + 1 ln Q 2 2 s s + 1 ln s + s + 1 0 which is the previous result already given in (4.2.20). The second example is the shifted Dirac function which has the following Laplace transform FðsÞ = Lfδðt − νÞg = e − ν s Introducing (4.2.31) into (4.2.29) we have

(4:2:31)

148

4 Mathematical Operations with Respect to the Order of the Bessel

(∞ ð

) δðλ − νÞ Jλ ðtÞ dλ

L

= LfJν ðtÞg =

e−ν

0

ln s +

pffiffiffiffiffiffiffiffi! s2 + 1

pffiffiffiffiffiffiffiffiffiffiffi s2 + 1

(4:2:32)

1 1 = pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν = pffiffiffiffiffiffiffiffiffiffiffi 2 + 1 Qν 2 2 s s + 1 ðs + s + 1Þ and (4.2.32) is identical with (4.2.14), the Laplace transform of the Bessel function of the first kind. In order to determinate higher derivatives with respect to the order, the chosen functions f(t) should be the derivatives of the shifted Dirac functions n o (4:2:33) FðsÞ = δðnÞ ðt − νÞ = ð − 1Þn sn e − ν s and using (4.2.29) we have (∞ ) ð L 0

n  ∂ Jν ðtÞ δðnÞ ðλ − νÞ Jλ ðtÞ dλ = L ∂νn  pffiffiffiffiffiffiffiffi h  pffiffiffiffiffiffiffiffiffiffiffi in s + s2 + 1 ln s + s2 + 1 n pffiffiffiffiffiffiffiffiffiffiffi = ð − 1Þ s2 + 1 h  pffiffiffiffiffiffiffiffiffiffiffi in n ln s + s2 + 1 n ðln QÞ ffi = ð − 1Þn pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi !ν = ð − 1Þ pffiffiffiffiffiffiffiffiffiffi s2 + 1 Q ν s2 + 1 s + s2 + 1 e − ν ln

n = 0, 1, 2, 3, . . . (4:2:34) Thus, formally from (4.2.34) it follows that the higher derivatives with respect to the order of the Bessel function of the first kind are given by determination of the Laplace transform inverses, i.e. by consecutive evaluation of complex integrals h pffiffiffiffiffiffiffiffiffiffiffi in c +ði ∞ s t n n e lnðs + s2 + 1Þ ∂ Jν ðtÞ ð− 1Þ = ffiffiffiffiffiffiffiffiffiffi ffi p p ffiffiffiffiffiffiffiffiffiffiffi ν ds 2πi ∂νn s2 + 1 ðs + s2 + 1Þ =

ð− 1Þ 2πi

n

c−i∞ c +ði ∞

c−i∞

es t ðln QÞn pffiffiffiffiffiffiffiffiffiffiffi ds s2 + 1 Q ν

n = 0, 1, 2, 3, . . .

;

Re s > c

In particular case of ν = 0, the expression (4.2.35) reduces to

(4:2:35)

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function



 ∂ n Jν ðtÞ ð − 1Þn = 2π i ∂νn ν=0 =

ð − 1Þn 2π i

h pffiffiffiffiffiffiffiffiffiffiffi in lnðs + s2 + 1Þ pffiffiffiffiffiffiffiffiffiffiffi ds s2 + 1

c +ði ∞ s t

e

c−i∞ c +ði ∞

c−i∞

149

(4:2:36)

es t ðln QÞn pffiffiffiffiffiffiffiffiffiffiffi ds s2 + 1

n = 0, 1, 2, 3, . . .

; Re s > c

Operational calculus permits expressing the first derivative with respect to the order ν differently than that was given by van der Pol in (4.2.8). The Laplace transform in (4.2.34) with n = 1 is pffiffiffiffiffiffiffiffiffiffiffi

 ∂ Jν ðtÞ lnðs + s2 + 1Þ = − pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffi ν ∂ν s2 + 1 ðs + s2 + 1Þ (3:2:37) pffiffiffiffiffiffiffiffiffiffiffi lnðs + s2 + 1Þ 1 pffiffiffiffiffiffiffiffiffiffiffi =− · pffiffiffiffiffiffiffiffiffiffiffi ν s2 + 1 ðs + s2 + 1Þ but [36] ( pffiffiffiffiffiffiffiffiffiffiffi ) lnðs + s2 + 1Þ π pffiffiffiffiffiffiffiffiffiffiffi L = − Y0 ðtÞ 2 s2 + 1 ( ) 1 ν −1 L pffiffiffiffiffiffiffiffiffiffiffi ν = Jν ðtÞ ; Re ν > 0 2 t ðs + s + 1 Þ −1

(4:2:38)

(4:2:39)

and from the convolution theorem we have ∂ Jν ðtÞ π ν Jν ðtÞ π ν = Y0 ðtÞ* = ∂ν 2 t 2

ðt Y0 ðt − xÞ Jν ðxÞ

dx x

0

(4:2:40)

Re ν > 0 The convolution integral can be converted to definite integral by introducing a new variable x = t (cos θ)2 ∂ Jν ðtÞ = πν ∂ν

π=2 ð

tan θ Y0 ½t ðsin θÞ2  Jν ½t ðcos θÞ2  dθ 0

(4:2:41)

Re ν > 0 The integrals in (4.2.41) can be written in a closed form only for ν = 1/2 and ν = 1 because derivatives with respect to the order are known. For other values of order ν, they should be evaluated numerically. If the same procedure is applied for n = 2, we have the product of three convolutions

150

4 Mathematical Operations with Respect to the Order of the Bessel

h pffiffiffiffiffiffiffiffiffiffiffi i2

2  lnðs + s2 + 1Þ ∂ Jν ðtÞ = pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffi ν ∂ν2 s2 + 1 ðs + s2 + 1Þ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi lnðs + s2 + 1Þ lnðs + s2 + 1Þ s pffiffiffiffiffiffiffiffiffiffiffi = · · pffiffiffiffiffiffiffiffiffiffiffi ν s s2 + 1 ðs + s2 + 1Þ

(4:2:42)

which is equivalent to

 ∂ 2 Jν ðtÞ π ν2 d Jν ðtÞ =− Y0 ðtÞ*Ji0 ðtÞ* 2 ∂ν2 dt t

(4:2:43)

Re ν > 0 However, if (4.2.42) is presented in different form h pffiffiffiffiffiffiffiffiffiffiffi i2

2  lnðs + s2 + 1Þ ∂ Jν ðtÞ = L ffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffi p ν ∂ν2 s2 + 1 ðs + s2 + 1Þ h pffiffiffiffiffiffiffiffiffiffiffi i2 lnðs + s2 + 1Þ s = · pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν 2 s s + 1 ðs + s2 + 1Þ and taking into account (4.2.32) and [36] 8h pffiffiffiffiffiffiffiffiffiffiffi i2 9 > = < lnðs + s2 + 1Þ > = − π Yi0 ðtÞ L− 1 > > s ; :

(4:2:44)

(4:2:45)

we have ∂ 2 Jν ðtÞ d = − π Yi0 ðtÞ* fJν ðtÞg ∂ν2 dt

(4:2:46)

Re ν > 0 or explicitly ðt ∂ 2 Jν ðtÞ π = − Yi0 ðt − xÞ ½Jν − 1 ðxÞ − Jν + 1 ðxÞ dx ∂ν2 2

(4:2:47)

0

In general case of the nth-derivative with respect to the order, the expression (4.2.34) is

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function

h pffiffiffiffiffiffiffiffiffiffiffi in

n  lnðs + s2 + 1Þ ∂ Jν ðtÞ n = ð − 1Þ pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffiffiffiffiffiffiffiffi ν ∂νn s2 + 1 ðs + s2 + 1Þ pffiffiffiffiffiffiffiffiffiffiffi !n lnðs + s2 + 1Þ sn = − · pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν s s2 + 1 ðs + s2 + 1Þ n = 0, 1, 2, 3, . . .

;

151

(4:2:48)

Re ν > − 1

and the compact form of inverse in (4.2.44) is ∂ n Jν ðtÞ = ð − 1Þn ½Ji0 ðtÞ*n *Jν ðnÞ ðtÞ ∂νn

(4:2:49)

n = 0, 1, 2, 3, . . . The above expression contains the product of n convolutions of the integral Bessel function of order zero and the nth derivative with respect to argument t of the Bessel function of the first kind of the order ν. The formula in (4.2.45) is inconvenient for numerical determination of derivatives with n > 2, but operational rules of the Laplace transformation permit to obtain behaviour of derivatives for small and large values of argument. For t → ∞, (4.2.48) becomes h pffiffiffiffiffiffiffiffiffiffiffi in

n  n lnðs + s2 + 1Þ ∂ Jν ðtÞ n n ½lnð1 + sÞ = ð − 1Þ L ffiffiffiffiffiffiffiffiffiffi ffi p pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ð − 1Þ ν n ∂ν ð1 + sÞ s2 + 1 ðs + s2 + 1Þ (4:2:50) n s ⁓ð − 1Þn ; s!0 ð1 + sÞν the inverse of (4.2.50) is [34] ∂ n Jν ðtÞ n! ν − n − 1 − t ðν − n − 1Þ = ð − 1Þn e Ln ðtÞ t ∂νn ΓðνÞ Re ν > 0

;

;

t!∞

(4:2:51)

n = 0, 1, 2, 3, . . .

where the generalized Laguerre polynomials are defined by the Rodrigues’ formula as −α LðαÞ n ðtÞ = t

et d n + α − t e  ½t n! dt

For t → 0, the approximation of (4.2.48) is h pffiffiffiffiffiffiffiffiffiffiffi in

n  n lnðs + s2 + 1Þ ∂ Jν ðtÞ n n 2 ½lnð2sÞ = ð − 1Þ pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ð − 1Þ ν + 1 n ∂ν ð2sÞ s2 + 1 ðs + s2 + 1Þ s!∞

(4:2:52)

(4:2:53)

152

4 Mathematical Operations with Respect to the Order of the Bessel

In tables of the Laplace transforms, the inverse of (4.2.53) is known only for n = 1 and n = 2 [36]  ν ( )     t 2 ∂ Jν ðtÞ 2 lnð2sÞ t − ψðν + 1Þ = ln = L− 1 − Γðν + 1Þ ∂ν 2 (4:2:54) ð2sÞν + 1 t!0 and ∂ 2 Jν ðtÞ = L− 1 ∂ν2

(

;

Re ν > − 1

2 ½lnð2sÞ2

)

ð2sÞν + 1 )  t ν (    2 t 2 − ψðν + 1Þ − ψ′ðν + 1Þ = ln Γðν + 1Þ 2 t!0

;

(4:2:55)

Re ν > − 1

Fényes [52, 76] considered the Laplace transforms of the moments of Bessel functions of the first kind (see Section 4.1)

(∞

)

ð

Fn ðsÞ = LfMn ðtÞg = L

νn Jν ðtÞ dν

(4:2:56)

0

t, s > 0 ;

n = 0, 1, 2, 3, . . .

where ∞ ð∞ ð

Fn ðsÞ =

e 0 0 ∞ ð

= 0

"∞ ð

∞ ð

− st n

ν Jν ðtÞ dν dt =

ν 0

νn

n

# e

− st

0

1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν dν = pffiffiffiffiffiffiffiffiffiffiffi 2 2 2 s +1 s + 1 ðs + s + 1Þ

Jν ðtÞ dt dν ∞ ð

νn e − ν ðs +

pffiffiffiffiffiffiffiffi

s2 + 1Þ

dν

(4:2:57)

0

n! n! ffi = pffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffi in + 1 = pffiffiffiffiffiffiffiffiffiffi 2 + 1 ð ln QÞn + 1 s 2 2 s + 1 lnðs + s + 1Þ and Fn + 1 ðsÞ =

ðn + 1Þ pffiffiffiffiffiffiffiffiffiffiffi Fn ðsÞ lnðs + s2 + 1Þ

(4:2:58)

By introducing f(t) = tn and its transform F(s) = n!/sn+1, the Laplace transform in (4.2.57) can directly be derived by using the operational rule given in (4.2.29). In the case n = 0, the Laplace transform was already discussed in (4.2.30). The limiting behaviour of moments Mn(t) for small and large values of t can be predicted using operational rules. The case of t → ∞, was already considered by Fényes [52] (see (4.2.29) and (4.2.30)). In operational treatment of the problem we have

4.2 Integration and Differentiation with Respect to the Order of the Bessel Function

n! n! n! Fn ðsÞ = pffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffi in + 1 ⁓ ½lnðs + 1Þn + 1 ⁓ sn + 1 s2 + 1 lnðs + s2 + 1Þ

153

(4:2:59)

s!0 and the inverse transform is

 n! L − 1 fFn ðs ! 0Þg = L − 1 n + 1 = tn s

;

t!∞

(4:2:60)

Thus, this approximation gives only the dominant power of t, (from polynomials determined by Fényes), but a very small value of the oscillatory terms in (4.1.29) are omitted. Fényes did not consider the case t → 0, but it is possible to show that for s → ∞ the Laplace transform can be approximated by n! 2 n! Fn ðsÞ = pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi n + 1 ⁓ 2 2 ð2 sÞ ½ lnð2 sÞn + 1 s + 1 ½ lnðs + s + 1Þ

; s!∞

and its inverse can be expressed in terms of the Volterra function ( )   2 n! t −1 −1 L fFn ðs ! ∞Þg = L =μ ,n 2 ð2sÞ½lnð2 sÞn + 1

(4:2:61)

(4:2:62)

t!0 where [30] ∞ ð

μðz, βÞ = 0

z x xβ dx Γðβ + 1Þ Γðx + 1Þ

and therefore, 1 Mn ðtÞ⁓ n!

∞ ð

0

 x t 2

xn

Γðx + 1Þ

dx

;

;

Re β > − 1

t!0

(4:2:63)

(4:2:64)

when this integral tends to zero if t → 0. As shown in Chapter 3, there is the substitution rule of the Laplace transformation which is connected with integrals or derivatives with respect to the order [30] ∞ ð FðsÞ = Lff ðtÞg = e − s t f ðtÞ dt

L

−1

0

∞  ð  ν + 1=2   pffiffiffi Jν + 1=2 ðtÞ 1 t 2 = π + 1Þ f ðνÞ dν F lnðs 2 Γðν + 1Þ ðs + 1Þ 2

(4:2:65)

0

This expression can be used if inverses of logarithmic functions are known. Unfortunately, they are rare, but the Dirac delta function or its derivatives are suitable

154

4 Mathematical Operations with Respect to the Order of the Bessel

for such operations. However, this leads to rather lengthy differentiations of integrands in (4.2.65), but the final results of such calculations are quite attractive. There is a similar substitution rule for the modified Bessel function [30] ∞ ð FðsÞ = Lff ðtÞg = e − s t f ðtÞ dt L

−1

0

∞  ð  ν + 1=2 pffiffiffi Iν + 1=2 ðtÞ 1 t 2 f ðνÞ dν − 1Þ = π F½lnðs Γðν + 1Þ ðs2 − 1Þ 2

(4:2:66)

0

and analogous expressions ∞ ð FðsÞ = Lff ðtÞg = e − s t f ðtÞ dt rffiffiffiffiffiffi ∞ ð  ν 2π t Jν − 1=2 ðtÞ f ðνÞ dν ΓðνÞ t 2 0 rffiffiffiffiffiffi ∞ ð  ν    2π t Iν − 1=2 ðtÞ −1 2 L F lnðs − 1Þ = f ðνÞ dν ΓðνÞ t 2 0

   L − 1 F lnðs2 + 1Þ =

(4:2:67)

0

4.3 Integration and Differentiation with Respect to the Order of the Modified Bessel Function of the First Kind by Applying the Laplace Transform Approach Laplace transforms of the Bessel functions of the second kind Yν(t) and the modified Bessel functions of the second kind Kν(t) exist only for a limited interval of ν values. This and their behaviour near the origin (the first function tends to minus infinity and the second function to plus infinity) makes the operational calculus less adaptable to use. On the other side, the modified Bessel function of first kind Iν(t) is interrelated with Jν(t) (the argument t is then replaced with it) and therefore the Laplace transform operations are very similar to those applied in Section 4.2. The modified Bessel function of the first kind has the following Laplace transform [36] pffiffiffiffiffiffiffiffiffi 2 e − ν lnðs + s − 1 Þ 1 1 pffiffiffiffiffiffiffiffiffiffiffi = pffiffiffiffiffiffiffiffiffiffiffi LfIν ðtÞg = pffiffiffiffiffiffiffiffiffiffiffi ν = pffiffiffiffiffiffiffiffiffiffiffi 2 2 s − 1 qν s −1 (4:3:1) s2 − 1 ðs + s2 − 1Þ pffiffiffiffiffiffiffiffiffiffiffi q = ðs + s2 − 1Þ ; Re s > 1 ; Re ν > − 1 Similarly, as in (4.2.13), let us evaluate the integral

4.3 Integration and Differentiation with Respect to the Order of the Modified Bessel

155

∞ ð

Iðν, tÞ =

Iλ ðtÞ dλ

(4:3:2)

ν

The integration under integral sign gives

(∞

)

ð

1 = pffiffiffiffiffiffiffiffiffiffiffi 2 s −1

Iλ ðtÞ dλ

LfIðν, tÞg = L ν

∞ ð

ν

dλ pffiffiffiffiffiffiffiffiffiffiffi λ ðs + s2 − 1Þ

(4:3:3)

1 = pffiffiffiffiffiffiffiffiffiffiffi s2 − 1 qν ln q The inverse of (4.3.3) is ∞ ð

Iðν, tÞ = ν

1 Iλ ðtÞ dλ = 2π i

c +ði ∞

c−i∞

est pffiffiffiffiffiffiffiffiffiffiffi ds ; 2 s − 1 qν ln q

Re s > 1

(4:3:4)

In this case, the Bromwich contour of complex integration, used in Figure 4.1 should be switched by the angle π/2 in the clockwise direction (Figure 4.2). The integrand in (4.3.4) has two branch points, but this time on the real axis, s = ± 1. The final result of integration is [4] y B

R→ ∞

C

D

J

I

–1

E

F

H

G

1

c

x

A

Figure 4.2: The Bromwich contour used in complex integration of equation (4.3.4).

156

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð

Iλ ðtÞ dλ = ν

ðπ

et 1 − 2 π 1 − π

∞ ð

0

et cos x sinðν xÞ

dx x

0

e − ν x − t cosh x ½x sinðπ νÞ + π cosðπ νÞ dx π 2 + x2

(4:3:5)

and in particular case of ν = 0 it reduces to ∞ ð

et Iλ ðtÞ dλ = − 2

0

∞ ð

0

e − t cosh x dx π 2 + x2

(4:3:6)

The asymptotic behaviour of integral (4.3.5) for t → 0 comes from 1 2 ⁓ LfIðν, t ! 0Þg = pffiffiffiffiffiffiffiffiffiffiffi ν+1 s2 − 1 qν ln q ð2sÞ lnð2sÞ

(4:3:7)

s!∞ which has the inverse in terms of the Volterra function [30] ( L− 1

2

ð2sÞν + 1 lnð2sÞ

)

 =ν



t ,ν = 2

∞ ð

0

 λ + ν t 2 dλ Γðλ + ν + 1Þ

(4:3:8)

t!0 and  ν t   t 2 Iðt, νÞ ν ,ν ⁓ Γðν + 1Þ lnð2=tÞ 2 ( ) ψðν + 1Þ ½ψðν + 1Þ2 − ψ′ðν + 1Þ − ... Iðt, νÞ = 1 − + lnð2=tÞ ½lnð2=tÞ2

(4:3:9)

t!0 In the case of t → ∞, it follows from (4.3.3) that 1 1 LfIðν, t ! ∞Þg = pffiffiffiffiffiffiffiffiffiffiffi ⁓ ν+1 2 2 ν s − 1 q ln q ðs − 1Þ and its inverse is [36] ( ) pffiffiffi ν + 1=2 1 πt −1 L Iν + 1=2 ðtÞ = 2ν + 1=2 Γðν + 1Þ ðs2 − 1Þν + 1

;

;

s!1

t!∞

(4:3:10)

(4:3:11)

157

4.3 Integration and Differentiation with Respect to the Order of the Modified Bessel

The exponential character of (4.3.11) is evident if the case ν = 0 is considered

 1 et −1 = sinh t⁓ L ; t!∞ (4:3:12) 2 ðs2 − 1Þ This is consistent with (4.3.6) where the contribution from the integral is negligible for large values of argument t. It is worthwhile to note, that the Laplace transform of (4.3.4) with the lower integration limit being zero, can also be derived by applying the operational rule analogous to (4.2.29)

(∞

)

ð

f ðλÞ Iλ ðtÞ dλ

L 0

h pffiffiffiffiffiffiffiffiffiffiffi i 1 1 = pffiffiffiffiffiffiffiffiffiffiffi F lnðs + s2 − 1Þ = pffiffiffiffiffiffiffiffiffiffiffi Fðln qÞ s2 − 1 s2 − 1

(4:3:13)

As expected, the differentiations of (4.3.5) with respect to the order gives the integral representation of the modified Bessel function of the first kind and its derivatives 1 Iν ðtÞ = π

ðπ e

t cos x

0

sinðνxÞ cosðν xÞ dx − π

∞ ð

e − ν x − t cosh x dx

(4:3:14)

0

and ∂ Iν ðtÞ 1 =− ∂ν π 1 + π

ðπ x et cos x sinðν xÞ dx 0 ∞ ð

(4:3:15) e − ν x − t cosh x ½x sinðπ νÞ − π cosðπ νÞ dx

0

In the case of ν = 0 we have ∞   ð ∂ Iν ðtÞ = − e − t cosh x dx = − K0 ðtÞ ∂ν ν = 0

(4:3:16)

0

Differentiation of (4.3.1) with respect to the order gives the Laplace transform pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffi

 − ν lnðs + s2 − 1 Þ e ln s + s2 − 1 ∂ Iν ðtÞ pffiffiffiffiffiffiffiffiffiffi =− L ∂ν s2 − 1  pffiffiffiffiffiffiffiffiffiffi (4:3:17) ln s + s2 − 1 ln q ; Re s > 1 ; Re ν > − 1 = pffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffi ν = pffiffiffiffiffiffiffiffiffiffi s2 − 1 qν s 2 − 1 s + s2 − 1 The function K0(t) is represented by the following inverse Laplace transform [36]

158

4 Mathematical Operations with Respect to the Order of the Bessel

8  pffiffiffiffiffiffiffiffiffiffiffi 9

> = ν < 1 L − 1  pffiffiffiffiffiffiffiffiffiffiffi ν = Iν ðtÞ > ; t : s + s2 − 1 >

(4:3:19)

Re ν > 0 the convolution theorem gives ∂ Iν ðtÞ ν = − K0 ðtÞ Iν ðtÞ = − ν ∂ν t

ðt K0 ðt − xÞ Iν ðxÞ

dx x

0

(4:3:20)

Re ν > 0 By introducing x = t (cos θ)2, the convolution integral (4.2.20) can be transformed to the following trigonometric integral ∂ Iν ðtÞ = − 2ν ∂ν

π=2 ð

tan θ K0 ½tðsin θÞ2  Iν ð½tðsin θÞ2 Þ dθ 0

(4:3:21)

Re ν > 0 Since the derivatives with respect to the order for ν = 1/2 and ν = 1 are known, (4.2.20) can be written also in the following forms − K0 ðtÞ

sinh t =− t3=2

ðt K0 ðt − xÞ

sinh x dx x3=2

(4:3:22)

1 = pffiffi ½et Eið − 2tÞ + e − t Eið2tÞ t and I1 ðtÞ K0 ðtÞ = t

ðt K0 ðt − xÞ

I1 ðxÞ I0 ðtÞ dx = − K1 ðtÞ x t

(4:3:23)

0

Similarly as in (4.2.29), the Laplace transforms of higher derivatives with respect to the order can be derived by introducing the shifted Dirac functions from (4.2.13) into (4.2.13)

4.3 Integration and Differentiation with Respect to the Order of the Modified Bessel

(∞ ð L 0

159

)

n  ∂ Iν ðtÞ δ ðλ − νÞ Iλ ðtÞ dλ = L ∂νn ðnÞ

pffiffiffiffiffiffiffiffi h 2

pffiffiffiffiffiffiffiffiffiffi in lnðs + s2 − 1 Þ pffiffiffiffiffiffiffiffiffiffi = ð − 1Þn s2 − 1 h pffiffiffiffiffiffiffiffiffiffi in lnðs + s2 − 1 Þ n = ð − 1Þ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ν s2 − 1 ðs + s2 − 1 Þ e − ν lnðs +

s −1Þ

ðln qÞn = ð − 1Þn pffiffiffiffiffiffiffiffiffiffi s2 − 1 q ν

; n = 0, 1, 2, 3, . . .

(4:3:24)

Re s > 1

and therefore higher derivatives with respect to the order are given as the Laplace transform inverses from h pffiffiffiffiffiffiffiffiffiffiffi in c +ði ∞ s t e lnðs + s2 − 1Þ ∂ n Iν ðtÞ ð − 1Þn = pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν ds 2π i ∂νn s2 + 1 ðs + s2 − 1Þ =

ð − 1Þn 2π i

c−i∞ c +ði ∞

c−i∞

(4:3:25)

es t ðln qÞn pffiffiffiffiffiffiffiffiffiffiffi ds s2 − 1 q ν

n = 0, 1, 2, 3, . . .

;

Re s > 1

For ν = 0 we have  n  ∂ Iν ðtÞ ð − 1Þn = n 2πi ∂ν ν=0 n

=

ð − 1Þ 2πi

c +ði ∞ s t

e

c−i∞ c +ði ∞

c−i∞

h pffiffiffiffiffiffiffiffiffiffiffi in lnðs + s2 − 1Þ pffiffiffiffiffiffiffiffiffiffiffi ds s2 − 1

es t ðln qÞn pffiffiffiffiffiffiffiffiffiffiffi ds ; s2 − 1

n = 0, 1, 2, 3, . . .

;

Re s > 1 (4:3:26)

The Laplace transform of the second derivative with respect to the order is h pffiffiffiffiffiffiffiffiffiffiffi i2

2  lnðs + s2 − 1Þ ∂ Iν ðtÞ = pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffi ν ∂ν2 s2 − 1 ðs + s2 − 1Þ h pffiffiffiffiffiffiffiffiffiffiffi i2 lnðs + s2 − 1Þ s = · pffiffiffiffiffiffiffiffiffiffiffi ν s ðs + s2 − 1Þ

(4:3:27)

160

4 Mathematical Operations with Respect to the Order of the Bessel

but

(h L− 1

pffiffiffiffiffiffiffiffiffiffiffi i2 lnðs + s2 − 1Þ

)

s

= Ki0 ðtÞ −

π2 4

(4:3:28)

and therefore from the convolution theorem we have   ∂2 Iν ðtÞ π2 ′ Iν ðtÞ = 2Ki ðtÞ − 0 4 ∂ν2 ′ Iν ðtÞ =

(4:3:29)

1 ½Iν − 1 ðtÞ + Iν + 1 ðtÞ 2

It is difficult to find an analogous formula to that in (4.2.49) for derivatives with respect to the order of the modified Bessel functions Iν(t). However, operational rules give at least the behaviour of derivatives for small and large values of argument t. For t → 0, both functions Jν(t) and Iν(t) behave similarly because (4.3.24) is identical with (4.2.53) for s → ∞ h pffiffiffiffiffiffiffiffiffiffiffi in

n  n lnðs + s2 − 1 Þ ∂ Iν ðtÞ n n 2 ½lnð2sÞ = ð − 1Þ L ffiffiffiffiffiffiffiffiffiffi ffi p pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ð − 1Þ ν + 1 n ∂ν (4:3:30) ð2sÞ s2 − 1 ðs + s2 − 1Þ s!∞ The Laplace transform inverses of (4.3.30) are known only for n = 1 and n = 2 (see (4.2.54) and (4.2.55)). For the first and second derivatives, with respect to the order, it is possible to obtain corresponding expressions of the Laplace transforms for s → 1 pffiffiffiffiffiffiffiffiffiffiffi

 ∂ Iν ðtÞ lnðs + s2 − 1Þ 1 1 = − pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ − pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ − ν + 1 2 2 2 ∂ν s s − 1 ðs + s − 1Þ s ðs + s − 1Þ (4:3:31) s!1 which has the simple inverse ∂ Iν ðtÞ tν ⁓− Γðν + 1Þ ∂ν

; t!∞

Re ν > 0 In the case of second derivative we have

(4:3:32)

4.3 Integration and Differentiation with Respect to the Order of the Modified Bessel

h pffiffiffiffiffiffiffiffiffiffiffi i2 pffiffiffiffiffiffiffiffiffiffiffi

2  lnðs + s2 − 1Þ ∂ Iν ðtÞ 2 lnðs + s2 − 1Þ 2 ln s = pffiffiffiffiffiffiffiffiffiffiffi L pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ pffiffiffiffiffiffiffiffiffiffiffi ν ⁓ ν + 1 ∂ν2 s s ðs + s2 − 1Þ s2 − 1 ðs + s2 − 1Þ

161

(4:3:33)

s!1 and ∂ 2 Iν ðtÞ 2 tν ⁓ ½ψðν + 1Þ − ln t Γðν + 1Þ ∂ν2

; t!∞

(4:3:34)

Re ν > 0 The Laplace transforms of moments of modified Bessel functions of the first kind were considered by Fényes in [52, 76] (∞ ) ð Gn ðsÞ = Lfmn ðtÞg = L

νn Iν ðtÞ dν

;

n = 0, 1, 2, 3, . . .

(4:3:35)

0

where ∞ ð∞ ð

Gn ðsÞ =

e 0 0 ∞ ð

= 0

"∞ ð

∞ ð

− st n

ν Iν ðtÞ dν dt =

ν 0

νn

n

# e

− st

0

1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ν dν = pffiffiffiffiffiffiffiffiffiffiffi 2 2 2 s −1 s − 1 ðs + s − 1Þ n!

Iν ðtÞ dt dν ∞ ð

νn e − ν ðs +

pffiffiffiffiffiffiffiffi

s 2 − 1Þ

dν

(4:3:36)

0

n! ffi = pffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffi in + 1 = pffiffiffiffiffiffiffiffiffiffi 2 − 1 ð ln qÞn + 1 s s2 − 1 lnðs + s2 − 1Þ

;

Re s > 1

with Gn + 1 ðsÞ =

ðn + 1Þ pffiffiffiffiffiffiffiffiffiffiffi Gn ðsÞ ln s + s2 − 1 

(4:3:37)

The expression (4.3.36) follows directly from (4.1.13) considering that for f(t) = tn we have F(s) = n!/sn+1. The Laplace transform in the case n = 0 was already discussed in (4.3.3). The limiting behaviour of moments mn(t) for large values of t was already considered by Fényes (see (4.1.31)). The Laplace transform of (4.3.36) for s → 1 can be approximated by

162

4 Mathematical Operations with Respect to the Order of the Bessel

n! Gn ðsÞ = pffiffiffiffiffiffiffiffiffiffiffih pffiffiffiffiffiffiffiffiffiffiffi i 2 s − 1 lnðs + s2 − 1Þ n + 1 ⁓

n! ðs2 − 1Þn=2 + 1

⁓

(4:3:38)

n!

;

2n=2 + 1 ðs − 1Þn=2 + 1

s!1

and therefore, ( −1

L fGn ðs ! 1Þg = L

−1

n!

)

2n=2 + 1 ðs − 1Þn=2 + 1

=

n! tn=2 et  Γ n2 + 1

2n=2 + 1

(4:3:39)

t!∞ This result is consistent with derived Fényes expressions in (4.1.31). In the case of small values of t, it is possible to show that the Laplace transform can be approximated by n! 2 n! Gn ðsÞ = pffiffiffiffiffiffiffiffiffiffiffih pffiffiffiffiffiffiffiffiffiffiffi in + 1 ⁓ ð2sÞ ½ lnð2 sÞn + 1 2 2 s − 1 lnðs + s − 1Þ

(4:3:40)

s!∞ which is identical with (4.2.61) and therefore ∞ ð  t x n 1 2 x mn ðtÞ⁓ dx ; t ! 0 n! Γðx + 1Þ

(4:3:41)

0

4.4 Differentiation with Respect to the Order of the Anger Function and the Weber Function by Applying the Laplace Transform Approach Since the Anger function is closely related to the Bessel function of the first kind, 1 Jν ðtÞ = π

ðπ 0

sinðπ νÞ cosðν x − t sin xÞ dx = Jν ðtÞ + π

∞ ð

e − t sinh x − ν x dx

(4:4:1)

0

an additional integration with respect to the argument t of the infinite integral in (4.4.1) gives the Laplace transform

4.4 Differentiation with Respect to the Order of the Anger Function

LfJν ðtÞg = LfJν ðtÞg +

sinðπ νÞ π

∞ ð

"∞ e − st

0

ð

163

# e − t sinh x − ν x dx dt

0

sinðπ νÞ = pffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiiν + π s 2 + 1 s + s2 + 1 1

∞ ð

0

(4:4:2) e− νx dx s + sinh x

;

Re s > 0

Introducing integral with respect to the order, similar to that in (4.2.13) ∞ ð

∞ ð

Jλ ðtÞ dλ =

Iðν, tÞ = ν

ν

1 Jλ ðtÞ dλ + π

"∞ ð

∞ ð

sinðπ νÞ ν

# e − t sinh x − ν x dx dν

(4:4:3)

0

and changing the order of integration in the second integral we have ∞ ð

Iðν, tÞ =

Jλ ðtÞ dλ = Iðν, tÞ ν

1 + π νð2

∞ ð

ν

e − t sinh x − ν x ½x sinðπ νÞ + π cosðπ νÞ dx π 2 + x2

(4:4:4)

Jλ ðtÞ dλ = Iðν2 , tÞ − Iðν1 , tÞ ν1

Combining (4.1.17) with (4.4.4), it follows that I(ν;t) becomes ∞ ð

Iðν, tÞ =

Jλ ðtÞ dλ = ν

1 1 + 2 π

ðπ sinðt sin x − νxÞ

dx x

(4:4:5)

0

and for ν = 0 it takes the form ∞ ð

Ið0, tÞ = 0

1 1 Jλ ðtÞ dλ = + 2 π

ðπ sinðt sin xÞ

dx x

(4:4:6)

0

The Laplace transform of this infinite integral is (∞ ) "∞ # ∞ ð ð − t sinh x ð e − st Jλ ðtÞ dλ = LfIð0, tÞg + e dx dt LfIð0, tÞg = L π 2 + x2 0

1

pffiffiffiffiffiffiffiffiffiffiffi + = pffiffiffiffiffiffiffiffiffiffiffi s2 + 1 lnðs + s2 + 1Þ

∞ ð

0

0

0

(4:4:7)

dx 2 2 ðπ + x Þðs + sinh xÞ

The asymptotic expressions of this integral for small and large argument t are [4]

164

4 Mathematical Operations with Respect to the Order of the Bessel

1 SiðπÞ Ið0, t ! 0Þ⁓ + t 2 π  2 3=2

; SiðπÞ = 1.85155... (4:4:8)

  π Ið0, t ! ∞Þ⁓1 + pffiffi sin t − 4 t π

The difference between the integrals (4.4.6) and (4.4.5) gives ! ! ðν ðπ 2 νx νx dx Jλ ðtÞ dλ = Ið0, tÞ − Iðν, tÞ = cos t sin x − sin π 2 2 x 0

(4:4:9)

0

It is worthwhile to compare behaviour of infinite integrals with respect to the order of the Bessel function of the first kind, the modified Bessel function of the first kind and the Anger function as a function of argument t. These three infinite integrals are plotted together in Figure 4.3. As can be observed, the oscillatory character of infinite integrals of the Bessel function of the first kind A and the Anger function C is nearly the same. The contribution coming from the integration of the second term in (4.4.1) is very small. Integrals A and C tend to unity as argument t tends to infinity. In the case of the modified Bessel function of the first kind, the exponential rise of infinite integral B is consistent with its behaviour predicted in (4.3.12). 2.0

B 1.5

C 1.0

A

0.5

0.0

0

2

4

t

6

8

10

Figure 4.3: Values of infinite integrals with respect to the order as a function of argument t of the Bessel function of the first kind A – equation (4.2.18); the modified Bessel function of the first kind B – equation (4.3.6); and the Anger function C – equation (4.4.6).

4.4 Differentiation with Respect to the Order of the Anger Function

165

Higher derivatives of the Anger function with respect to the order can easily be derived if (4.4.1) is written in complex form 1 Jν ðtÞ = Jν ðtÞ + π

∞ ð

e − t sinh x ½e − ðx − π iÞ ν − e − ðx + π iÞ ν  dx 2i

(4:4:10)

0

and therefore, ∂n Jν ðtÞ ∂n Jν ðtÞ ð − 1Þn = + 2 πi ∂νn ∂νn

∞ ð

e − t sinh x − ν x ½ðx − π iÞn eπ i ν − ðx + π iÞn e − π i ν  dx

0

(4:4:11) For ν = 0, the expression (4.4.11) becomes 

∂n J ν ðtÞ ∂νn

 ν=0

=

∞  n  ð ∂ Jν ðtÞ ð − 1Þn + e − t sinh x ½ðx − π iÞn − ðx + π iÞn  dx 2 πi ∂νn ν = 0 0

(4:4:12) From (4.4.12), the first derivative with respect to the order for ν = 0 is ∞     ð ∂J ν ðtÞ ∂Jν ðtÞ = + e − t sinh x dx ∂ν ν = 0 ∂ν ν = 0

(4:4:13)

0

and taking into account that the limiting expression of the Bessel and Anger functions is available, it is possible to present (4.4.13) in the following form π π H0 ðtÞ = Y0 ðtÞ + 2 2

∞ ð

e 0

− t sinh x

1 dx = π

π=2 ð

sinðt sin xÞ dx

(4:4:14)

0

Similarly, the second derivative from (4.4.12) is ∞  2   2  ð ∂ J ν ðtÞ ∂ Jν ðtÞ = − 2 x e − t sinh x dx 2 ∂ν2 ∂ν ν=0 ν=0 0

(4:4:15)

166

4 Mathematical Operations with Respect to the Order of the Bessel

where 

∂2 J ν ðtÞ ∂ν2

(

 ν=0

=

π2 − + ðln t + γ + ln 2Þ 2

)

2 − 2 ðln t + γ + ln 2Þ

2 + π ðln t + γ + ln 2Þ Y0 ðtÞ + π

2

J0 ðtÞ

h i2 ð1 lnð1 − x2 Þ pffiffiffiffiffiffiffiffiffiffiffi cosðt xÞ dx 1 − x2

(4:4:16)

0

An analogous definition of moments, as in (4.1.22) and (4.1.23) for the Anger function is ∞ ð

M n ðtÞ =

∞ ð

ν Jν ðtÞ dν = 0

t>0 ;

∞ ð

ν Jν ðtÞ dν +

n

n

0

"

sinðπ νÞ ν π

0

#

∞ ð

n

e

− t sinh x − ν x

dx dν

0

(4:4:17)

n = 0.1.2.3....

Changing the order of integration we have "∞ # ∞ ð ð 1 − t sinh x −νx n e e ν sinðπ νÞ dν dx Mn ðtÞ = Mn ðtÞ + π 0

(4:4:18)

0

where Mn(t) moments are given in (4.1.24). The inner integral in (4.1.26) can be treated as a known Laplace transform, and therefore, ∞ ð n! e − t sinh x gn ðxÞ dx M n ðtÞ = Mn ðtÞ + π 0   n d π ; n = 0, 1, 2, 3, ... gn ðxÞ = ð − 1Þn n d x ðπ2 + x2 Þ

(4:4:19)

and n! ð − 1Þn n! LfMn ðtÞg = pffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiin + 1 + π s2 + 1 s + s2 + 1

∞ ð

0

gn ðxÞ dx ðs + sinh xÞ

(4:4:20)

The contribution coming from the integral in (4.4.20) is small, and therefore the expected asymptotic behaviour of the Anger function moments resembles those of the Bessel function of the first kind.

4.4 Differentiation with Respect to the Order of the Anger Function

167

The Weber function represented by the following expression: 1 Eν ðtÞ = − Yν ðtÞ − π

∞ ð

e − t sinh x ½eν x + e − ν x cosðπ νÞ dx

;

t>0

(4:4:21)

0

is less suitable for the operational treatment because the Bessel function of second kind Yν(t) has the Laplace transform only for limited interval of the orders ν. However the Anger and Weber functions are interrelated in the following way [18] 1 π

J ν ðtÞ ± i Eν ðtÞ =

ðπ

e ± i ðν x − t sinh xÞ dx

(4:4:22)

0

and therefore, 1 Eν ðtÞ = π 1 = π

ðπ sinðνx − t sinh xÞ dx 0 ðπ

(4:4:23) ½sinðνxÞ cosðt sinh xÞ − cosðνxÞ sinðt sinh xÞ dx

0

The Laplace transform of the Weber function from (4.4.23) is "∞ # ð ðπ 1 − st LfEν ðtÞg = sinðνxÞ e cosðt sinh xÞ dt dx π 0

1 − π =

0

"∞ ð

ðπ cosðνxÞ 0

ðπ

1 π

0

# e

− st

sinðt sinh xÞ dt dx

(4:4:24)

0

s sinðν xÞ − cosðν xÞ sinh x s2 + ðsinh xÞ2

dx

and the corresponding moments are ∞ ð

mn ðtÞ =

1 ν Eν ðtÞ dν = π

0

1 − π

ðπ

t>0

"∞ ð

ðπ

n

0

"∞ ð

#

ν cosðπ νÞ dν dx n

sinðt sinh xÞ 0

ν sinðπ νÞ dν dx

cosðt sinh xÞ 0

# n

(4:4:25)

0

; n = 0.1.2.3....

The inner integrals in (4.4.25), as suggested by Fényes, can be evaluated using the Laplace transforms of trigonometric functions

168

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð

gn ðsÞ =

e 0 ∞ ð

hn ðsÞ =

e

dn π ν sinðπ νÞ dν = ð − 1Þ d sn π 2 + s2

− sν n

!

n

dn ν cosðπ νÞ dν = ð − 1Þ d sn

− sν n

s π 2 + s2

n

0

!

(4:4:26)

n = 0.1.2.3.... In the limit, by putting s → 0, we have  gn ðs ! 0Þ = ð − 1Þn lim s!0

 ∞ ð dn  π νn sinðπ νÞ dν = dsn π2 + s2 0

g2n + 1 ðs ! 0Þ = 0 

 ∞ ð dn  s hn ðs ! 0Þ = ð − 1Þ lim νn cosðπ νÞ dν = s!0 dsn π 2 + s2

(4:4:27)

n

0

h2n ðs ! 0Þ = 0

4.5 Integration and Differentiation with Respect to the Order of the Struve Functions by Applying the Laplace Transform Approach Derivatives with respect to the order of the Struve function can be derived starting from the Laplace transform-inverse pair [36] n L t

ν=2

∞ pffiffi o ð − s t ν=2 pffiffi Hν ðz tÞ = e t Hν ðz tÞ dt 0

  i  z ν − z2 =4 s iz p ffiffiffi e erf =− s 2s 2 s Re ν > −

3 2

;

(4:5:1)

Re s > 0

Differentiation of the left side of (4.5.1) with respect to ν gives pffiffi pffiffi o tν=2 pffiffi ∂ n ν=2 ν=2 ∂Hν ðz t Þ ln t Hν ðz tÞ + t t Hν ðz tÞ = 2 ∂ν ∂ν and of the Laplace transform of (4.5.1) is

(4:5:2)

4.5 Integration and Differentiation with Respect to the Order of the Struve Functions

   ln z n n o o  ν pffiffi ∂ z iz 2 2s e − z =4 s erf pffiffiffi · L tν=2 Hν ðz tÞ = − i s ∂ν 2s 2 s Taking into account the following inverses

   pffiffi z ν − z2 =4 s iz z −1 i = − tðν − 1Þ=2 Hν − 1 ðz tÞ e erf pffiffiffi L s 2s 2 2 s

 ln s = − ðγ + ln tÞ L− 1 s

169

(4:5:3)

(4:5:4)

from the convolution theorem we have t pffiffi o z ð ðν − 1Þ=2 pffiffi ∂ n ν=2 x lnðt − xÞ Hν − 1 ðz tÞ dx t Hν ðz tÞ = ∂ν 2 0

zγ + 2

ðt x

ðν − 1Þ=2

z pffiffi pffiffi Hν − 1 ðz tÞ dx + tν=2 ln Hν ðz tÞ 2

(4:5:5)

0

Considering that ðt

xν Hν ðxÞ dx = tν Hν ðtÞ

(4:5:6)

0

and introducing t = 1 into (4.5.2) and (4.5.5), the derivative with respect to the order of the Struve function is ∂ Hν ðz Þ h z i = ln + γ Hν ðzÞ + z ∂ν 2

ð1 0

xν lnð1 − x2 Þ Hν − 1 ðz xÞ dx (4:5:7)

1 Re ν > − 2 For some values of ν, the derivatives with respect to the order are known and therefore the integral in (4.5.7) can be evaluated. The behaviour of the inverse Laplace transforms in (4.5.3) for small and large values of z gives the asymptotic formulas for the considered derivatives (for ν = 0 and ν = 1 see [5]). The Laplace transformation technique also permits to evaluate the following integral with respect to the order ðν Iðz, t, νÞ = 0

pffiffi tλ=2 Hλ ðz tÞ dλ

(4:5:8)

170

4 Mathematical Operations with Respect to the Order of the Bessel

by using the Laplace transform in (4.5.1) !ν #  " 2 ie − z =4s iz z   erf pffiffiffi −1 LfIðz, t, νÞg = 2s 2s 2 s s ln z

(4:5:9)

The inverse of (4.5.9) can be determined by applying the convolution theorem in several ways, by using relevant transform-inverse pairs [36] !) ( z 1 − ν pffiffi i − z2 =4 s iz −1 = L erf pffiffiffi tðν − 1Þ=2 Hν − 1 ðz tÞ − νe s 2 2 s !) ( pffiffi i iz sinðz tÞ 2 L − 1 − pffiffiffi e − z =4 s erf pffiffiffi = pffiffiffiffiffi s 2 s πt (4:5:10) ( )    1 1 t −1 L ,α = ν a a ðs aÞα + 1 lnðs aÞ ∞ ð

νðt, αÞ = 0

tx + α dx Γðx + α + 1Þ

;

a>0

;

Re α > − 1

Thus, the desired integral can be expressed in terms of the associated Volterra functions ν(t,α) [30] as ðν

pffiffi tλ=2 Hλ ðz tÞ dλ

0

rffiffiffiffiffiffiffiffi

    pffiffi z zt 1 zt 1 −ν = sinðz tÞ ν ,− ,ν− 2πt 2 2 2 2

(4:5:11)

or ðν

pffiffi tλ=2 Hλ ðz tÞ dλ

0

pffiffi    pffiffi pffiffi z zt 2 − π H1 ðz tÞ ðν − 1Þ=2 pffiffi  −t Hν − 1 ðz tÞ = sinðz tÞ ν 2 2 π t

(4:5:12)

In particular case, if the upper limit of integration is equal to infinity, the explicit form of (4.5.11) is ∞ ð

t 0

λ=2

pffiffi Hλ ðz tÞ dλ =

pffiffiffiffiffiffiffiffiffi  rffiffiffiffiffiffi ðt  z sinðz t − xÞ z x 1 pffiffiffiffiffiffiffiffiffi ν dx ,− 2π 2 2 t−x 0

(4:5:13)

4.5 Integration and Differentiation with Respect to the Order of the Struve Functions

171

For t = 1 and x = 1 – w2, (4.5.13) becomes ∞ ð

0

rffiffiffiffiffi ð1   2z z 1 2 Hλ ðzÞ dλ = sinðz wÞ ν ð1 − w Þ, − dw π 2 2 0

ð1 =z

H − 1 ðz wÞ ν

hz i ð1 − w2 Þ dw 2

(4:5:14)

0

2 H − 1 ðz wÞ = − H1 ðz wÞ π The above integral can also be written as trigonometric integral by introducing w = cosθ ∞ ð

0

rffiffiffiffiffi π=2   ð 2z z 1 2 sin θ dθ Hλ ðzÞ dλ = sinðz cos θÞ ν ðsin θÞ , − π 2 2

(4:5:15)

0

Using the Struve function expansion (2.9.9) z 2 k+μ+1 2  ð − 1Þ  Hμ ðzÞ = Γ k + 32 Γ k + μ + 32 k=0 ∞ X

k

(4:5:16)

and changing the order of integration and summation, the integrals with respect to the order in (4.5.13) and (4.5.14) can be developed in the series of the Volterra functions ∞ ð

t 0

λ=2

pffiffi Hλ ðz tÞ dλ =

rffiffiffiffiffi ∞ z t  k 1 2z X k z ν 2 ,k+ 2 ð − 1Þ 1 · 3 · 5 · . . . ð2k + 1Þ π k=0

(4:5:17)

and ∞ ð

0

rffiffiffiffiffi ∞   zk ν z2 , k + 21 2z X Hλ ðz Þ dλ = ð − 1Þk 1 · 3 · 5 · . . . ð2k + 1Þ π k=0

(4:5:18)

Derivatives and integrals with respect to the order of the modified Struve function Lν(z) can be determined in a similar way as demonstrated for Hν(z), both functions are interrelated

172

4 Mathematical Operations with Respect to the Order of the Bessel

Lν ðzÞ = − i e − π i ν=2 Hν ðizÞ

(4:5:19)

and therefore, ∂ Lν ðzÞ πi ∂ Hν ði zÞ =− Lν ðzÞ − ie − π i ν=2 ∂ν 2 ∂ν

(4:5:20)

Using (4.5.7) we have ð1 ∂ Lν ðz Þ h z i = ln + γ Lν ðzÞ + z xν lnð1 − x2 Þ Lν − 1 ðz xÞ dx ∂ν 2 0

Re ν > −

(4:5:21)

1 2

The recurrence relation between the modified Struve functions z ν 2ν Lν − 1 ðzÞ − Lν + 1 ðzÞ = Lν ðzÞ + pffiffiffi 2 z π Γ ν + 32

(4:5:22)

gives ∂ Lν − 1 ðzÞ ∂ Lν + 1 ðzÞ 2ν ∂ Lν ðzÞ 2 − = + Lν ðzÞ ∂ν ∂ν z ∂ν z z ν     z 3 2  ln + −ψ ν+ pffiffiffi 3 2 2 πΓ ν+ 2

(4:5:23)

The derivative with respect to the of the modified Struve function can be also presented in the following way [30]     ∂ Lν ðzÞ z 1 Lν ðzÞ = ln −ψ ν+ ∂ν 2 2  ν     2 z2 z 3  p ffiffiffi Iðν, zÞ ln + −ψ ν+ 2 2 π Γ ν + 21 ð1 Iðν, zÞ =

ð1 − x2 Þ 0

Using (4.5.21) and

ν − 1=2

lnð1 − x2 Þ sinhðz xÞ dx

(4:5:24)

173

4.5 Integration and Differentiation with Respect to the Order of the Struve Functions

CiðizÞ =

1 πi ½EiðzÞ − E1 ðzÞ + 2 2

i ½EiðzÞ + E1 ðzÞ 2 rffiffiffiffiffiffi 2 sinh z L − 1=2 ðzÞ = πz rffiffiffiffiffiffi 2 L1=2 ðzÞ = ðcosh z − 1Þ πz

Siði zÞ =

(4:5:25)

it is possible to obtain the derivative with respect to the order for ν = − 1/2 rffiffiffiffiffiffi    ∂ Lν ðzÞ 2 e− z ez = ½ Eið2 zÞ − 2EiðzÞ + ½E1 ð2 zÞ − 2E1 ðzÞ 2 2 ∂ν πz ν = − 1=2 and for ν = 1/2   ∂ Lν ðzÞ ∂ν ν = 1=2 ) rffiffiffiffiffiffi ( h z i 2 e− z ez = ½ Eið2 zÞ − 2EiðzÞ − ½ E1 ð2 zÞ − 2E1 ðzÞ − γ + ln 2 2 πz 2

(4:5:26)

(4:5:27)

The last expression (4.5.27) permits to evaluate the integral in (4.5.24) for ν = 1/2 ð1

z i 1h γ + ln cosh z z 2

z  e e− z ½ E1 ð2 zÞ − 2E1 ðzÞ − ½Eið2 zÞ − 2EiðzÞ + 2 2

lnð1 − x2 Þ sinhðz xÞ dx = − 0

(4:5:28) Starting in the same way as in (4.5.1), the order derivatives of the modified Struve function can be derived by using the following the Laplace transform [36]   n pffiffi o 1  z ν z2 =4 s z e erf pffiffiffi L tν=2 Lν ðz tÞ = − s 2s 2 s (4:5:29) 3 Re ν > − ; Re s > 0 2 The analogous expression to that in (4.5.7) is

174

4 Mathematical Operations with Respect to the Order of the Bessel

ð1 ∂ Lν ðz Þ h z i = ln + γ Lν ðzÞ + z xν lnð1 − x2 Þ Lν − 1 ðz xÞ dx ∂ν 2 0

(4:5:30)

1 Re ν > − 2 and to that in (4.5.11) is ðν Iðz, t, νÞ =

pffiffi tλ=2 Lλ ðz tÞ dλ

0

rffiffiffiffiffiffiffiffi

    pffiffi z zt 1 zt 1 −ν = sinhðz tÞ* ν ,− ,ν− 2π t 2 2 2 2

(4:5:31)

If the upper limit of integration is equal to infinity and t = 1, the explicit form of (4.5.31) is ∞ ð

0

rffiffiffiffiffi ð1   2z z 1 Lλ ðzÞ dλ = sinhðz wÞ ν ð1 − w2 Þ, − dw π 2 2

(4:5:32)

0

Integrals with respect to the order in (4.5.31) can be developed in the series of the associated Volterra functions from the modified Struve function expansion (2.9.10) z 2 k + μ + 1  2  Lμ ðzÞ = Γ k + 32 Γ k + μ + 32 k=0 ∞ X

(4:5:33)

The inversion of the order of integration and summation in (4.5.33) gives ∞ ð

t 0

λ=2

pffiffi Lλ ðz tÞ dλ =

rffiffiffiffiffi ∞   2 z X zk ν z2t , k + 21 π k = 0 1 · 3 · 5 · . . . ð2k + 1Þ

(4:5:34)

and ∞ ð

0

rffiffiffiffiffi ∞   2 z X zk ν z2 , k + 21 Lλ ðz Þ dλ = π k = 0 1 · 3 · 5 · . . . ð2k + 1Þ

(4:5:35)

The asymptotic behaviour of the order derivatives or integrals of the modified Struve function as a function of argument z is discussed for a number of particular values of ν in [5]. The Laplace transform technique, which is used to obtain

4.5 Integration and Differentiation with Respect to the Order of the Struve Functions

175

derivatives and integrals with respect to the order of the Struve functions (see (4.5.1) and (4.5.30)) can easily be extended to the Bessel functions because their Laplace transforms are similar, even in simpler form [36] n pffiffi o 1 L tν=2 Jν ðz tÞ = s n o pffiffi 1 L tν=2 Iν ðz tÞ = s

 z ν 2s  z ν 2s

e− z ez

2 =4 s

2 =4 s

; ;

Re s > 0 Re s > 0

(4:5:36) (4:5:37)

Differentiation of (4.5.36) and (4.5.37) with respect to the order gives pffiffi o 1  z ν − z2 =4 s  z ∂ n ν=2 e ln L t Jν ðz tÞ = ∂ν s 2s 2s

      ν − 1 1 z z 2s 2 − e − z =4 s = ln s 2s 2s z

(4:5:38)

pffiffi o 1  z ν z2 =4 s  z ∂ n ν=2 e ln L t Iν ðz tÞ = ∂ν s 2s 2s

      1 z ν − 1 z2 =4 s z 2s − e = ln s 2s 2s z

(4:5:39)

and

Using (4.5.7), (4.5.36) and (4.5.37), derivatives with respect to the order of the Bessel functions are given in terms of the following convolution integrals    pffiffi o z ðν − 1Þ=2 pffiffi ∂ n ν=2 zt (4:5:40) Jν − 1 ðz tÞ γ + ln t Jν ðz tÞ = t ∂ν 2 2    pffiffi o z ðν − 1Þ=2 pffiffi ∂ n ν=2 zt (4:5:41) Iν − 1 ðz tÞ γ + ln t Iν ðz tÞ = t ∂ν 2 2 If the same procedure is extended to integrals with respect to the order of the Bessel functions, then we have (ðν ) # "  z ν pffiffi 1 2 =4 s λ=2 − z  e L t Jλ ðz tÞ dλ = −1 (4:5:42) 2s s ln 2zs 0

(ðν ) pffiffi λ=2 L t Iλ ðz tÞ dλ = 0

# "  z ν 1 2 =4 s z  e −1 2s s ln 2zs

(4:5:43)

and taking into account (4.5.10), the logarithmic term will lead to the convolution integrals including the Volterra and Bessel functions.

176

4 Mathematical Operations with Respect to the Order of the Bessel

ðν

pffiffi tλ=2 Jλ ðz tÞ dλ =

rffiffiffiffiffi     pffiffi pffiffi z zt 1 z zt − tðν − 1Þ=2 Jν − 1 ðz tÞν J − 1=2 ðz tÞ ν ,− 2t 2 2 2 2

0

ðν

(4:5:44) rffiffiffiffiffi     pffiffi pffiffi pffiffi z zt 1 z zt − tðν − 1Þ=2 Iν − 1 ðz tÞν tλ=2 Iλ ðz tÞ dλ = I − 1=2 ðz tÞ ν ,− 2t 2 2 2 2

0

(4:5:45) Thus, expressions of derivatives and integrals with respect to the order in terms of convolution integrals are based on the product theorem and manipulations with Laplace transforms of the Bessel and related functions. These convolution integrals can be presented in different forms because they depend on how Laplace transforms are arranged in the products for inversion processes.

4.6 Differentiation with Respect to the Order of the Kelvin Functions by Applying the Laplace Transform Approach Functions introduced by Kelvin, berν(z) and beiν(z), represent the real part of the Bessel function Jν(z) of order ν and of argument i3/2z and the imaginary part of the Bessel function Jν(z) of order ν and of argument i −3/2z respectively. They can be treated by using the corresponding formulas which are applied to the Bessel function of first kind. However, the Kelvin functions are usually investigated separately, because they occur in a special type of mathematical problems arising in the theory of electrical current, fluid mechanics and elasticity. Similarity of the Kelvin functions with the Bessel functions is evident when operations with transforms and inverses in the Laplace transformation are performed. The Laplace transform of the Kelvin function of the first kind is [36]  2  n pffiffi o 1  z ν z 3π ν cos + L tν=2 berν ðz tÞ = 4s s 2s 4 (4:6:1) Re ν > − 1

; Re s > 0

and its derivative with respect to the order is

4.6 Differentiation with Respect to the Order of the Kelvin Functions

pffiffi 

ν=2 ∂ ½ t berν ðz tÞ L ∂ν pffiffi

 pffiffi pffiffi ν=2 ∂ berν ðz tÞ =L t + tν=2 lnð tÞ berν ðz tÞ ∂ν  2   2    ν 1 z z z 3π ν 3 π  z ν z 3π ν − ln sin + + = cos 4s 4s s 2s 2s 4 4s 2 s 4 The inverse of the second term in (4.6.2) is known

  2  pffiffi 3π z ν z 3π ν 3 π ν=2 = sin + t beiν ðz tÞ L−1 4s 4s 2 s 4 4

177

(4:6:2)

(4:6:3)

and taking into account (4.6.1) and (4.5.4), the final result which is based on the convolution theorem is pffiffi z pffiffi pffiffi ∂ ½ tν=2 berν ðz tÞ 3π ν=2 =− t beiν ðz tÞ + tν=2 ln berν ðz tÞ ∂ν 4 2 pffiffiffi ðt n pffiffi pffiffi o 2z xðν − 1Þ=2 ½lnðt − xÞ + γ berν − 1 ðz tÞ + beiν − 1 ðz tÞ dx − 4 0

Re ν > 0 (4:6:4) Introducing t = 1 and changing value of parameter z, the expression for the derivative with respect to the order of the Kelvin function of the first kind becomes pffiffiffi   pffiffiffi pffiffiffi ∂ berν ðz 2 Þ 3π z =− beiν ðz 2 Þ + ln pffiffiffi berν ðz 2Þ ∂ν 4 2 1 ð n pffiffiffiffiffi pffiffiffiffiffi o z − xðν − 1Þ=2 ½lnð1 − xÞ + γ berν − 1 ðz 2 xÞ + beiν − 1 ðz 2x Þ dx 2 0

Re ν > 0 (4:6:5) In the case of ν = 0, the Laplace transform should be treated in a different way to obtain inversion of the product  2  2 ln s z ln s 1 z = pffiffiffi · pffiffiffi cos (4:6:6) cos 4s 4s s s s From

178

4 Mathematical Operations with Respect to the Order of the Bessel

 ln s 1 pffiffiffi = − pffiffiffiffiffi ðln t + γ + 2 ln 2Þ L s πt rffiffiffi! rffiffiffi!

 2  1 z 1 t t −1 pffiffiffi cos = pffiffiffiffiffi cos z L cosh z 4s 2 2 s πt −1

the final result after few more steps is pffiffiffi     pffiffiffi pffiffiffi ∂ berν ð 2 zÞ 3π z =− beiðz 2 Þ + ln pffiffiffi berðz 2Þ ∂ν 4 2 ν=0 ð1

1 + π

0

pffiffiffi pffiffiffi ½lnð1 − xÞ + γ + 2 ln 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðz xÞ coshðz xÞ dx xð1 − xÞ

(4:6:7)

(4:6:8)

Changing the integration variable in (4.6.8), it is possible to obtain the convolution integral as the finite trigonometric integral pffiffiffi     pffiffiffi pffiffiffi ∂ berν ð 2 zÞ 3π z =− beiðz 2 Þ + ln pffiffiffi berðz 2Þ ∂ν 4 2 ν=0 4 + π

π=2 ð

pffiffiffi pffiffiffi lnð2C sin θÞ cosðz 2 cos θÞ coshðz 2 cos θÞ dθ

0

γ = 2 ln C

; C = 1.3346682...

ð4:6:9Þ

Following the same procedure, the equivalent expression for the Kelvin function of the second kind is pffiffiffi   pffiffiffi pffiffiffi ∂ beiν ðz 2 Þ 3π z = berν ðz 2 Þ + ln pffiffiffi beiν ðz 2Þ ∂ν 4 2 ð1 n pffiffiffiffiffi pffiffiffiffiffi o z − xðν − 1Þ=2 ½lnð1 − xÞ + γ berν − 1 ðz 2 xÞ − beiν − 1 ðz 2x Þ dx 2 0

Re ν > 0 (4:6:10) and for ν = 0 we have pffiffiffi     pffiffiffi pffiffiffi ∂ beiν ð 2 zÞ 3π z = berðz 2 Þ + ln pffiffiffi beiðz 2Þ ∂ν 4 2 ν=0 1 + π

ð1 0

pffiffiffi pffiffiffi ½lnð1 − xÞ + γ + 2 ln 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðz xÞ sinhðz xÞ dx xð1 − xÞ

(4:6:11)

179

4.7 Differentiation with Respect to the Order of the Integral Bessel Functions

or in the form of the trigonometric integral pffiffiffi     pffiffiffi pffiffiffi ∂ beiν ð 2 zÞ 3π z = berðz 2 Þ + ln pffiffiffi beiðz 2Þ ∂ν 4 2 ν=0 4 + π

π=2 ð

pffiffiffi pffiffiffi lnð2C sin θÞ sinðz 2 cos θÞ sinhðz 2 cos θÞ dθ

0

γ = 2 ln C

;

C = 1.3346682... (4:6:12)

4.7 Differentiation with Respect to the Order of the Integral Bessel Functions by Applying the Laplace Transform Approach The integral Bessel function of the first kind is defined in (2.1.68) as ∞ ð

Jiν ðtÞ = −

Jν ðxÞ dx x

(4:7:1)

t

and the Laplace transform of it resembles that of the Bessel function ( ) 1 1 LfJiν ðtÞg = pffiffiffiffiffiffiffiffiffiffiffi − 1 ν s ðs + s2 + 1 Þν

(4:7:2)

Re ν > 0 If the complex inversion formula is applied to (4.7.2) 1 Jiν ðtÞ = 2πiν

c +ði ∞

c−i∞

" # est 1 pffiffiffiffiffiffiffiffiffiffiffi − 1 ds s ðs + s2 + 1 Þν

(4:7:3)

Re s > c the integral representation of the integral Bessel function is derived by using the Bromwich contour plotted in Figure 4.1. The result of complex integration in (4.7.3) is [4] Jiν ðtÞ = −

1 1 − 2ν π ν

sinðπ νÞ + πν

ðπ cot x sinðt sin x − ν xÞ dx 0 ∞ ð

0

(4:7:4) e − t sinh x − ν x coth x dx

180

4 Mathematical Operations with Respect to the Order of the Bessel

The equivalent form of integral representation of Jiν(t) can be derived starting with the Schlaefli representation of Jν(t) 1 π

Jν ðtÞ =

ðπ cosðt sin x − νxÞ dx − 0

sinðπ νÞ π

∞ ð

e − t sinh x − ν x dx

0

(4:7:5)

Re ν > − 1 The integral Bessel function is reached by an integration of (4.7.5) # ∞ ∞ ð ð "ðπ Jν ðuÞ 1 du cosðu sin x − νxÞ dx Jiν ðtÞ = − du = − u π u t t 0 " # ∞ ð ð ∞ sinðπ νÞ du e − u sinh x − ν x dx + π u t

(4:7:6)

0

Changing the order of integration in (4.7.6) and taking into account that ∞ ð

siðtÞ = −

sin x dx x

t ∞ ð

CiðtÞ = − ∞ ð

E1 ðtÞ =

cos x dx x

(4:7:7)

t

e− x dx x

t

we have 1 − cosðπ νÞ sinðπ νÞ Jiν ðtÞ = + 2ν π 1 + π

ðπ

∞ ð

e − ν x E1 ðt sinh xÞ dx

0

(4:7:8)

½cosðνxÞ Ciðt sin xÞ + sinðν xÞ siðt sin xÞ dx 0

For ν = 0, the expected result is obtained by applying l’Hôpital’s rule Ji0 ðtÞ =

1 π

ðπ Ciðt sin xÞ dx

(4:7:9)

0

If the order ν is odd or even positive integer, the expression (4.7.8) can be represented by ðπ 1 1 sin½ð2n + 1Þx siðt sin xÞ dx (4:7:10) + Ji2 n + 1 ðtÞ = 2n + 1 π 0

4.7 Differentiation with Respect to the Order of the Integral Bessel Functions

181

because ðπ cos½ð2n + 1Þx Ciðt sin xÞ dx = 0 0

(4:7:11)

n = 0, 1, 2, 3, ... and 1 Ji2 n ðtÞ = π

ðπ sinð2n xÞ Ciðt sin xÞ dx

(4:7:12)

0

because ðπ sinð2nxÞ siðt sin xÞ dx = 0 0

(4:7:13)

n = 1, 2, 3, ... These results can be written differently, if odd and even values of ν are introduced into (4.7.4) 1 Ji2 n + 1 ðtÞ = − 2ð2 n + 1Þ (4:7:14) ðπ 1 cot x cos½ð2n + 1Þ x sinðt sin xÞ dx − ð2 n + 1Þ π 0

because ðπ cot x sin½ð2n + 1Þx cosðt sin xÞ dx = 0 0

(4:7:15)

n = 0, 1, 2, 3, ... and Ji2 n ðtÞ = −

1 1 + 4n 2 nπ

ðπ cot x sinð2 nxÞ cosðt sin xÞ dx

(4:7:16)

0

because ðπ cot x cosð2n xÞ sinðt sin xÞ dx = 0 0

n = 1, 2, 3, ...

(4:7:17)

182

4 Mathematical Operations with Respect to the Order of the Bessel

Equalities of involved integrals can be established if (4.7.10) is compared with (4.7.14) and (4.7.12) with (4.7.16). Direct differentiation with respect to the order of the integral representation in (4.7.8) gives ∂ Jiν ðtÞ 1 − cosðπ νÞ π sinðπ νÞ − = ∂ν 2ν2 2ν ∞ ð sinðπ νÞ e − ν x E1 ðt sinh xÞ ½x sinðπ νÞ − π cosðπ νÞ dx + π 1 + π

(4:7:18)

0

ðπ

x ½sinðνxÞ Ciðt sin xÞ − cosðν xÞ siðt sin xÞ dx 0

For ν = 0, by applying l’Hôpital’s rule we have ∞   ð ðπ ∂ Jiν ðtÞ π2 1 − = x siðt sin xÞ dx + E1 ðt sinh xÞ dx + 4 ∂ν π ν=0 0

(4:7:19)

0

The first derivative with respect to the order the integral Bessel function can be derived also by using the Laplace transform method. From (4.7.2) it follows that pffiffiffiffiffiffiffiffiffiffiffi − ν pffiffiffiffiffiffiffiffiffiffiffi )

 ( ∂ Jiν ðtÞ 1 − ½s + s2 + 1  ln½s + s2 + 1  = − L pffiffiffiffiffiffiffiffiffiffiffi ν ν2 s ∂ν νs ½s + s2 + 1  (4:7:20) Re ν > 0 and using (4.7.3) it is possible to present the inverse of (4.7.20) as ∂ Jiν ðtÞ Jiν ðtÞ 1 = − ∂ν ν 2πiν Re s > c

c +ði ∞

c−i∞

pffiffiffiffiffiffiffiffiffiffiffi e s t lnðs + s2 + 1Þ pffiffiffiffiffiffiffiffiffiffiffi ds s ðs + s2 + 1 Þν

(4:7:21)

; Re ν > 0

Solving the complex integral with the help of Bromwich contour (Figure 4.1), the final result of the inverse is [4] ∂ Jiν ðtÞ Jiν ðtÞ 1 = + ∂ν ν πν −

1 πν

∞ ð

0

Re ν > 0

ðπ x cot x cosðt sin x − ν xÞ dx 0

e − t sinh x − ν x coth x ½ x sinðπ νÞ − π cosðπ νÞ  dx

(4:7:22)

4.8 Zeros of the Bessel and Related Functions As a Function of Their Order

183

This integral representation can be present in a more compact form taking into account the inverse Laplace transforms [36] ( pffiffiffiffiffiffiffiffiffiffiffi ) 2 − 1 lnðs + s + 1Þ L = Ji0 ðtÞ s (4:7:23) ( ) 1 ν Jν ðtÞ −1 pffiffiffiffiffiffiffiffiffiffiffi = L t ðs + s2 + 1Þ which permit to express the integral in (4.7.21) as the convolution integral ∂ Jiν ðtÞ Jiν ðtÞ = − ∂ν ν

ðt Ji0 ðt − xÞ Jν ðxÞ

dx x

0

(4:7:24)

Re ν > 0 or by changing the variable of integration we have the finite trigonometric integral ∂ Jiν ðtÞ Jiν ðtÞ = −2 ∂ν ν

π=2 ð

tan θ Ji0 ½t ðsin θÞ2  Jν ½t ðsin θÞ2  dθ 0

(4:7:25)

Re ν > 0

4.8 Zeros of the Bessel and Related Functions As a Function of Their Order In two investigations performed in the period 1918–1919, Watson [77] examined transmission of electrical waves around the Earth. For the first time, as a solution of this physical problem he found the necessity to determine zeros (roots) of functional expressions (algebraic sums of products) when the order of the Bessel and Hankel functions is a variable. In 1936, as the next step, Coulomb [47] postulated that in the case of the Bessel function of the first kind, if argument z is real and positive, its zeros are real, simple and asymptotically close to the negative integers for large nth zeros. About two, three decades later, in various physical and technological studies (quantum-mechanical potential scattering and in numerous diffraction phenomena), similar transcendental functions were encountered. The unknown quantities to be determined were series of orders ν1(z), ν2(z), ν3(z), ν4(z) . . . which were associated with prescribed values of arguments z. Values of these orders were physically identified as poles of scattering amplitudes resulting from interaction of various kinds of waves with spheres and cylinders. Thus, various physical or engineering problems, in different geometric situations, were linked with determination of zeros of the Bessel and

184

4 Mathematical Operations with Respect to the Order of the Bessel

related functions as a function of their order. And therefore, this topic found some kind of importance in the theory of electromagnetic waves and also in the solution of non-self-adjoint boundary value problems. The most significant results from this subject are reported briefly here by using a uniform notation. They are presented in the final form, without giving derivations and proofs. In the mathematical literature [61, 64–68, 90–95], three equations, with an increasing complexity, were solved. The first from these equations is Jν ðzÞ = 0 Jνn ðzÞ = 0

,

(4:8:1)

n = 1, 2, 3, ...

where the argument z is fixed. Using the asymptotic expansions of the Bessel function for particular values of argument z and order ν, Cohen [64] solved (4.8.1) and he found that for 0 < | z | ≪ 1, and/or | ν | ≫ | z | + 1 z 2 n

 2 n + 2  2 n + 4 2 n 2 n z2 z e − +O νn ðzÞ = − n + n!ðn − 1Þ! ðn − 1Þ ðn − 1Þ! ðn + 1Þ! n2 n + 1 2

(4:8:2)

and this result is consistent with the Coulomb prediction [47]. If order and argument have similar value z ~ ν, but | z – ν | is large, then νn ðzÞ = z −

 qn z1=3 3 q2n − +O z− 1 1=3 2=3 1=3 6 10 6 z

(4:8:3)

Aiðqn Þ = 0 where qn are zeros of the Airy function and they are tabulated in [9]. If order is considerably smaller than argument | z | ≫ | ν | and | arg(z) | < π, the nth zero of (4.8.1) is νn ðzÞ = ν0 ðzÞ +

 ½4 ν0 ðzÞ2 − 1 ½4 ν0 ðzÞ2 − 1½4 ν0 ðzÞ2 − 25 +O z− 5 + 4π z 192 π z3

ν0 ðzÞ = − 2 n −

1 2z + 2 π

;

n≈

(4:8:4)

z π

In the next equation to solve, the derivative of the Bessel function with regard to the argument z is considered d Jν ðzÞ =0 dz

(4:8:5)

Cohen found that solutions of (4.8.5) are similar to those derived for the solution in (4.8.1). For 0 < | z | ≪ 1, and /or | ν | ≫ | z | + 1, his result was

4.8 Zeros of the Bessel and Related Functions As a Function of Their Order

z 2 n νn ðzÞ = − n −

2

n!ðn − 1Þ!

+

2 ðn2 − 2Þ

 z 2 n + 2 2

ðn2 − 1Þðn!Þ2

+O

 2 n + 4 2 n z e n2 n + 1

In the case of z ~ ν, | z – ν | is large, he found that h i 2 3 ðq′n Þ − 2′  q′n z1=3 qn − +O z− 1 νn ðzÞ = z − 1=3 2=3 1=3 10 6 z 6

185

(4:8:6)

(4:8:7)

Ai′ðq′n Þ = 0 where the zeros of derivatives of the Airy function are considered to be known. For | z | ≫ | ν | and | arg(z) | < π, the expression is νn ðzÞ = ν0 ðzÞ +

 4 ν0 ðzÞ2 + 3 16 ν0 ðzÞ4 + 181 ν0 ðzÞ2 − 63 +O z− 5 + 3 4π z 192 π z

1 2z ν0 ðzÞ = − 2 n − + 2 π

;

(4:8:8)

z n≈ π

In the third equation to solve, the Bessel function and its derivative with regard to argument z appear in the form d Jν ðzÞ + α Jν ðzÞ = 0 dz

(4:8:9)

where α is given real or complex constant. This type of equation frequently emerges in a solution of boundary value problems. Cohen obtained, for this case that if 0 < | z | ≪ 1 and/or | ν | ≫ | z | + 1, the following expression for the zeros of equation (4.8.9) z 2 n νn ðzÞ = − n −

2

n!ðn − 1Þ!

−

4α

z 2 n + 2 2

ðn!Þ2

+O

 2 n + 4 2 n z e n2 n + 1

(4:8:10)

jνj > jαj If z ~ ν and | z – ν | is large, the solution is represented by h i ′n Þ 2 − 2 3 ðq 1=3 ′  q z q′n νn ðzÞ = z − n1=3 − +O z− 1 2=3 1=3 10 6 z 6  z 1=3 Aiðq′n Þ = 0 Ai′ðq′n Þ + α 6

(4:8:11)

In the case | z | ≫ | ν | and | arg(z) | < π, Cohen found two expressions valid for small and large values of parameter α

186

4 Mathematical Operations with Respect to the Order of the Bessel

νn ðzÞ = ν0 ðzÞ − −

ðα2 − 1Þ ½4 ν0 ðzÞ2 + 3 + α2 ðα2 − 1Þ ½4 ν0 ðzÞ2 − 1 4π z

3αð1 − α2 Þ½4 ν0 ðzÞ2 − 1½4 ν0 ðzÞ2 + 15 − 96π z2 2

 2αf½4 ν0 ðzÞ2 + 3 + α2 ½4 ν0 ðzÞ2 − 1g +O z− 3 − 96 π z2  α2 2α 1 − 3 1 2z z ν0 ðzÞ = − 2 n − + ; n≈ − ; jαj  1 π 2 π π

(4:8:12)

and νn ðzÞ = ν0 ðzÞ − −

ð1 − 3α2 Þ f½4 ν0 ðzÞ2 + 3 + α2 ½4 ν0 ðzÞ2 − 1g 12 π z α4

α2 ð1 + 3α2 Þ½4 ν0 ðzÞ2 − 1½4 ν0 ðzÞ2 + 15 − 96 π z2 α5

 ½4 ν0 ðzÞ2 + 3 + α2 ½4 ν0 ðzÞ2 − 1 +O z− 3 2 5 96π z α  1 1 2 − 3 α 1 2z z 3α ν0 ðzÞ = − 2 n + + ; n≈ − ; π 2 π π

(4:8:13)

−

jαj  1

Starting from the mid-fifties of the previous century, a number of investigations were directed to finding zeros of the Hankel functions, the derivatives with respect to the argument z and the linear combination of them all as a function of the order [91–95]. The simplest equations to solve are Hνð1Þ ðzÞ = 0

;

Hνð2Þ ðzÞ = 0

ðzÞ = 0 Hνð1Þ n

;

n = 1, 2, 3, ...

(4:8:14)

However, in this case, Magnus and Kotin [68] observed that only one Hankel function should be taken into account, because for a real argument, zeros of these functions are conjugates and the Hankel functions are interrelated in the following way ð1Þ

Hν* ðzÞ = Hνð2Þ ðz*Þ ð1Þ

ð2Þ

Hν ðxÞ = Hν* ðxÞ

;

x>0 (4:8:15)

ð1Þ

H − ν* ðxÞ = − Hνð1Þ ðeπ i xÞ π ν i ð1Þ Hν ðzÞ H ð1Þ − ν ðzÞ = e

z = x + iy

; z* = x − iy

;

ν = α + iβ

; ν* = α − iβ

4.8 Zeros of the Bessel and Related Functions As a Function of Their Order

187

where a bar denotes the complex conjugate of the indicated Hankel function. The Magnus and Kotin paper is important because it includes detailed analysis about the number of zeros and their location as a function of complex and real arguments. Magnus and Kotin evaluated only large zeros which are practically important. They demonstrated that if 0 < argν < π, and ν is sufficiently large, the real and imaginary parts of the large zeros of the Hankel function of the first kind are    π  3  πð4n − 1Þ  − 2 ln −θ n−  ð1 + εn Þ 2 4  2e r     1  πð4n − 1Þ  − 1 ln βn = π n −  ð1 + δn Þ 4  2e r   lnðln nÞ  ; δn ⁓εn ; n  1 εn = O ln n 

αn = π

(4:8:16)

In this expression, some misprints which were mentioned in [66] are corrected. Magnus and Kotin [68] found that for real, positive large argument x, a much simpler relation exists νn ðxÞ⁓x +

   x1=3 3 2=3 π i=3 e 3π n + 2 4

;

x1

(4:8:17)

Following the Cohen method which was used in determination of zeros of the Bessel function of the first kind, Keller, Rubinow and Goldstein [66] derived corresponding expressions for specific ranges of the order and argument. For 0 < | z | ≪ 1 or | ν | ≫ 1 + | z |2, their result for zeros of the equation (4.8.14) is π ni  Fðr, θÞ ln 2r  π   π 2 i 2 −θ −γ i 2 −θ −γ  Fðr, θÞ = 1 + +  r 2 ln 2r ln 2 νn ðzÞ = −

 π 3 i 2 −θ −γ − +   r  3 ln 2 z = rei θ

ζ ð3Þ π2 n2 3

 +O

z2 lnðz=2Þ

(4:8:18) 

; j zj  1

In the case of z fixed and ν large, the expression (4.8.16) slightly differs by a numerical coefficient from that given below

188

4 Mathematical Operations with Respect to the Order of the Bessel

      π  1  πð4n − 1Þ  − 2 lnðln nÞ ln 1 + O −θ n−  2 4  2e r ln n     − 1    1  πð4n − 1Þ  lnðln nÞ ln ; n1 1+O βn = π n −   4 2e r ln n αn = π

(4:8:19)

If the absolute values of the argument and order are large, Keller, Rubinow and Goldstein [66] quoted the Franz [79] result νn ðzÞ = z +

 qn z1=3 eπ i=3 61=3 ðqn Þ2 e2 π i=3 + +O z− 1 ; 61=3 180 z1=3

j zj  n > 0

(4:8:20)

Aiðqn Þ = 0 Zeros of the derivatives with respect to the argument of the Hankel function of the first kind are symmetric about the origin, and therefore only solutions of ð1Þ

d Hν ðzÞ =0 dz

(4:8:21)

are needed for real values of ν and different ranges of the argument z. For small values of z, the final result resembles (4.8.18)  π i n − 21  Fðr, θÞ νn ðzÞ = − ln 2r  π   π 2 i 2 −θ −γ i 2 −θ −γ r Fðr, θÞ = 1 + +   r 2 ln 2 ln 2 (4:8:22)  π 3 ζ ð3Þ π2 n2   2 i 2 −θ −γ − z 3 + +O   r 3 lnðz=2Þ ln 2

iθ

z = re

; j zj  1

In case of | ν | ≫ 1 + | z |2, the real and imaginary components of ν, which are associated with (4.8.21) are    − 2    π  3  πð4n − 3Þ  lnðln nÞ ln 1 + O −θ n−  2 4  2 er ln n     − 1    3  πð4n − 3Þ  lnðln nÞ ln ; n1 1+O βn = π n −   4 2e r ln n αn = π

(4:8:23)

4.8 Zeros of the Bessel and Related Functions As a Function of Their Order

For | z | large and n fixed, the derivative of the Airy function is involved " # 2  q′n z1=3 eπ i=3 61=3 e2 π i=3 ðq′n Þ 1 νn ðzÞ = z + + + +O z− 1 1=3 1=3 180 6 z 10q′n Ai′ðq′n Þ = 0

189

(4:8:24)

; j zj  n > 0

In addition, Keller, Rubinow and Goldstein [66] considered the linear combination of (4.8.14) and (4.8.21) ð1Þ

d Hν ðzÞ + iα Hνð1Þ ðzÞ = 0 dz

(4:8:25)

However, their results are less convenient for numerical determinations, e.g. for | ν | ≫ 1 + | z |2 or | z | ≪ 1, they were able to give zeros of (4.8.25) only as a solution of the rather complicated transcendental equation h i + νn ðzÞ    2  ln Γð1 π i ð21 − nÞ Γð1 − νn ðzÞ αz z +O (4:8:26) νn ðzÞ = + +O  z π i  z π i νn ðzÞ νn ðzÞ 2 ln 2 − 2 ln 2 − 2 The solution of (4.8.25) for | z | large and n fixed was also derived by Levy and Keller [67] νn ðzÞ = z +

 qn ðαz1=3 Þ z1=3 − 1=3 + O z 61=3

e5π i=6 α z1=3 lptAi′ðqn Þ − Aiðqn Þ = 0 61=3

(4:8:27)

; j zj  n > 0

The importance of determination of zeros with respect to the order of the Hankel function, its derivative and the linear combination of them led Cochran, [65] in 1965, to reconsider solutions of (4.8.14), (4.8.21) and (4.8.25). His attention was directed to large values of the order ν and unrestricted values of argument z. In order to obtain solution for such conditions, Cochran derived an asymptotic expansion of the Hankel function of the first kind directly in terms of the Airy functions and not as usually in terms of the Bessel functions. He postulated that for large | z |, the ν – zeros of equation Hνð1Þ ðzÞ = 0 are all negative real numbers, and they can be approximated by

(4:8:28)

190

4 Mathematical Operations with Respect to the Order of the Bessel

νn ðzÞ = z +

qn z1=3 e − 2 π i=3 1 + 1=3 Oð1Þ 1=3 2 z

; j zj  1

Aiðqn Þ = 0     3 π n 2=3 1 1 + Oð1Þ qn ≈ − 2 n

(4:8:29)

and for fixed z, the large order zeros of the Hankel function can be evaluated from π ni lnðln nÞ   ½1 + Oð1Þ ; νn ðzÞ =  π 3π n ln n i − arg z + ln 2 ejzj

n1

(4:8:30)

In the case of the ν – zeros of derivatives of the Hankel function of the first kind ð1Þ

d Hν ðzÞ =0 dz

(4:8:31)

Cochran found for large | z | that " # z 1=3 − 4 π i=3 ′n Þ 2 e ðq 1 1 − 2 π i=3 q′n e + + Oð1Þ − νn ðzÞ = z + 30 2 z 5 q′n ð2zÞ1=3

(4:8:32)

j zj >> 1 and for large ν " # 2 z 2 w=3 ′ − 2 π i=3 ½lnðq′n Þ qne 1+ Oð1Þ ; νn ðzÞ = e 3 2 ðq′n Þ

jnj > > 1

3=2 2ðq′n Þ e − π i ez   1 q′n = qn 1 + Oð1Þ n

w e2 w=3 =

(4:8:33)

The result given in (4.8.33) is also nearly valid for solution of ð1Þ

d Hν ðzÞ + iα Hνð1Þ ðzÞ = 0 dz

(4:8:34)

when large ν – zeros of the Hankel function are considered. For large values of the argument z, Cochran derived the following expression:

4.8 Zeros of the Bessel and Related Functions As a Function of Their Order

2 z 1=3 ðq′n Þ e − 4 π i=3 1 − 2 π i=3 ′ νn ðzÞ = z + qne + 2=3 + Oð1Þ ; 2 z 2 30 z1=3 z 1=3 e − π i=6 Aiðq′n Þ = 0 Ai′ðq′n Þ − α 2

q′n = qn +

191

jzj > > 1 (4:8:35)

21=3 eπ i=6 1 + 2=3 Oð1Þ α z1=3 z

He also gave the estimated accuracy of large order zeros when evaluated by his approximated expressions. From studies devoted to determination of the order zeros, it is worthwhile to mention also two papers of Streifer and Kodis [70, 71]. They investigated diffraction effects in cylindrical geometry by using the Hankel functions of the first kind. However, the solution of physical problem involved significantly more complicated equations, in the form of the following quotients: ! ! ð1Þ ð2Þ d Hν ðzÞ d Hν ðwÞ z w dz dw − = 0 ; z, w > > 1 (4:8:36) ð1Þ ð2Þ Hν ðzÞ Hν ðwÞ and ! ð1Þ d Hν ðzÞ z dz ð1Þ

Hν ðzÞ

  d Jν ðwÞ w dw − =0 Jν ðwÞ

(4:8:37)

where z and w denote prescribed positive values of arguments. Solutions for particular values of z and w are very long and complex to be presented here, they are available in the Streifer and Kodis papers [24, 71, 72]. From mathematical point of view, the most elaborate are two investigations performed by Nagase [96, 97] in 1954. He studied the diffraction of elastic waves by a sphere and found the necessity to determine zeros with respect to the order of the linear homogenous forms with the Hankel functions and their derivatives f1 ðλ, zÞ Hνð2Þ ðzÞ + f2 ðλ, zÞ λ=ν

ð2Þ

dHν ðzÞ =0 dz

(4:8:38)

2

where the argument z is real positive number and the order ν is a complex variable. In the prescribed functions f1 and f2, it was convenient to use λ = ν 2 and not ν as a variable. Nagase gave general expressions for estimation of the zeros of (4.8.38) by using asymptotic expansions of the Hankel functions in forms which are valid for a

192

4 Mathematical Operations with Respect to the Order of the Bessel

concerned region of argument. He devoted much attention to the case when the argument z and the order ν are nearly the same. Nagase also involved the Nicolson integral for an estimation of his solutions which were expressed in the form of infinite integrals. He discussed in a detail two particular cases of (4.3.38) which are important in mathematical physics ð2Þ

α ð2Þ dHν ðzÞ H ðzÞ + =0 z ν dz

(4:8:39)

where α is a real constant, and the following bilinear form f1 ðλ, z, αÞ Hνð2Þ ðzÞ Hνð2Þ ðα zÞ + f2 ðλ, z, αÞ + f3 ðλ, z, αÞ Hνð2Þ ðzÞ + f4 ðλ, z, αÞ α −1

yν, 1 < yν + 1, 1 < yν, 2 < yν + 1, 2 < yν, 3 < yν + 1, 3 < .... ;

ν> −1

(4:9:3) (4:9:4)

In the case of zeros of functions and derivatives it is possible to write the following inequality sequences: ν ≥ j′ν, 1 < yν, 1 < y′ν, 1 < jν , 1 < j′ν, 2 < yν , 2 < y′ν , 2 < iν, 2 < jν, 2 < j′ν, 3 < ....

(4:9:5)

j′ν, 1 < j′ν + 1, 1 < j′ν, 2 < j′ν + 1, 2 < j′ν, 3 < j′ν + 1, 3 < ....

(4:9:6)

y′ν, 1 < y′ν + 1, 1 < y′ν, 2 < y′ν + 1, 2 < y′ν, 3 < y′ν + 1, 3 < ....

(4:9:7)

jν + 1, k < j′ν , k + 1

;

k = 1, 2, 3, ...

(4:9:8)

jν + 1, k < j′ν , k j′′ν , k > j′ν , k j′′ν , k > jν , k

; ;

00

(4:9:14)

0

e − 2 ν t K0 ð2 jν, k sinh tÞ dt

; ν>0

(4:9:15)

0 ∞ P

jν, k ½Jν + k ðjν, k Þ2 d jν, k k = 1 = P ∞ dν ðν + kÞ ½Jν + k ðjν, k Þ2 k=1

(4:9:16)

196

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð d j′ν, k 2 jν, k e − 2 ν t ½ðj′ν, k Þ2 coshð2tÞ − ν2  K0 ð2j′ν, k sinh tÞdt = 2 dν 2 ′ ½ðj ν, k Þ − ν  0

d j′′ν, k dν

j′ν, k ≠ jνj

1  Ð 2 2 ′′ ν ½Jν ðj′′ν, k xÞ dx − ½J ðj Þ ν ν, k x 0  = 1 Ð 2 2 ′′ ′′ ′′ j ν, k x ½Jν ðj ν, k xÞ dx − ½Jν ðj ν, k Þ 0

d j′′ν, k = dν

2ν

( j′′ Ðν, k 0

− ½Jν ðj′′ν, k Þ

2

∞ ð

; ν>0

(4:9:18)

ν>0

(4:9:19)

) 2 ½Jν ðj′′ν, k Þ dx x

ðj′′ν, k Þ Jν ðj′′ν, k Þ J ′′′ν ðj′′ν, k Þ

d cν, k = 2cν, k dν

(4:9:17)

2

;

e − 2 ν t K0 ð2 cν, k sinh tÞ dt

(4:9:20)

0 ∞ ð d cν, k 2c′ν, k 2 e − 2 ν t ½ðc′ν, k Þ coshð2 tÞ − ν2 K0 ð2c′ν, k sinh tÞ dt = dν ðc′ν, k − ν2 Þ

(4:9:21)

0

These formulas help to establish properties of the zeros of the Bessel functions and their derivatives as a function of the order ν. This permits to establish whether on investigated interval of ν, the function increases, decreases, is monotonic, convex or concave. Starting from Watson [7], there is a very large number of inequalities for estimation of the zeros of the Bessel functions and their derivatives. In the case of first smallest zeros, the upper bounds are:   ν 3 1 ; ν> + (4:9:22) jν, 1 < π 2 4 2 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi (4:9:23) jν, 1 < ν + 1 ½ ν + 2 + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jν1 < ðj0, 1 Þ2 + 2ν½π2 − ðj0, 1 Þ2  ; 0 < ν < (4:9:24) 2 jν, 1 < j0, 1 +

πν 2

;

ν>0

(4:9:25)

and the lower bounds are: jν, 1 > π ðν + 1Þ

;

−1 ν2 − 19 + 6 2ν2 + 10 ν + 17 ;

(4:9:27) (4:9:28) ν>0

(4:9:29)

ðν + 1Þ ðν2 + 12 ν + 23Þ pffiffiffiffiffiffiffiffiffiffi ; ν>0 ν+4

1=4 pffiffiffiffiffiffiffiffiffi 2 jν, 1 > ν + 1 1 + ðν + 1Þ ; ν≥ −1 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jν, 1 > ðν + 1Þðν + 5Þ ; − 1 < ν < ∞  π 1 jν, 1 > j0, 1 + 2ν j0, 1 − ; −

jν1 > j0, 1 + 2 νðπ − j0, 1 Þ jν1 >

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðj0, 1 Þ2 + ν2 ;

; 00

(4:9:36)

and in particular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi νðν + 2Þ < jν, 1 < 2 ðν + 1Þð ν + 3Þ ; 0 < ν < 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðν + 1Þð ν + 5Þ ; 1 − 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 ðν + 1Þ < jν, 1 < ν + 1ð1 + ν + 2Þ ; ν > 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðν + 1Þ ðν2 + 12ν + 13Þ 2ðν + 1Þð ν + 5Þð5ν + 1Þ < jν, 1 < ν+4 7 ν + 19

(4:9:37) (4:9:38)

(4:9:39) (4:9:40) (4:9:41) ; ν≥0

(4:9:42)

The first three zeros of Jν(z) and Yν(z) as a function of ν, for large values of the order can be expressed by:

198

4 Mathematical Operations with Respect to the Order of the Bessel

jν, 1 ⁓ν + 1.85575708 ν1=3 + −

1.0331502 0.00397406 − ν ν1=3

0.0907627 0.0433385 + + ... ; ν5=3 ν7=3

jν, 2 ⁓ν + 3.2446076ν1=3 +

(4:9:43)

ν1

3.958244 0.08331 0.8437 − − 5=3 + ... ν ν1=3 ν

(4:9:44)

5.759713 0.22607 2.8039 − − 5=3 + ... ν ν1=3 ν

(4:9:45)

ν1 jν, 3 ⁓ν + 4.3816712 ν1=3 + ν1 and it is known also that " # β 7β2 49 β3 78363β4 jν, 1 = 2 β 1 + − + − + ... 1152 276480 4 96

;

−10

k = 1, 2, 3, ...

 1=3 ν 1=3 ν 1=3 3 2 < jν, k < ν + ak + ðak Þ2 2 2 20 ν

Aið− ak Þ = 0

(4:9:52)

;

ν>0

(4:9:53)

; k = 1, 2, 3, ...

ν  jν, k <  jak j 1−

; ν>

3π k 23=2 (4:9:54)

21=3 ν2=3

Aið − ak Þ = 0 ; jν, k
ν +

2 jak − 1 j3=2 3

; kν ;

ν≥

1 2

Aið− ak Þ = 0 ; k = 1, 2, 3...  2=3  2=3 3π 3π < jak j < ð4k − 1.4Þ ð4k − 0.965Þ 8 8

(4:9:56)

200

4 Mathematical Operations with Respect to the Order of the Bessel

−

  2=3  2=3 3π 3 5 3π < ak < − ð4k − 1Þ + tan − 1 ð4k − 1Þ 8 2 18πð4k − 1Þ 8

δk ≤ jν, k ≤ δk − jν, k ≤ δk

;

4 ν2 − 1 8δk

ν≥

1 2

;

;

ν 1 − Þ 2 4

δk = π ðk +

;

0 −1

2 ðν + 1Þ ðν + 5Þ ð5ν + 1Þ 7ν + 19

ðjν1 Þ2 < ðj0, 1 Þ2 ð1 + νÞ

;

;

(4:9:58) ν> −1

(4:9:59)

−1 4ðν + 1Þ ν + 2 ðjν, 1 Þ2 > ðν + 1Þðν + 5Þ ðjν, 1 Þ2 < 2 ðν + 1Þ ðν + 3Þ ðjν, 1 Þ2 > ðν + 1Þ ðν +

ðjν, 1 Þ2 >

13 Þ 2

(4:9:64) ;

ν> −1 ;

(4:9:65)

ν> −1

(4:9:66)

; ν> −1

(4:9:67)

24 ðν + 1Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2ðν2 − 1Þ 1 − 2ν + ð2ν + 1Þð2ν + 3Þ

ðjν, 1 Þ2 > ðν + 1Þ ðν +

(4:9:62)

2π2 1 − Þ 3 2

;

ν>

;

ν> −1

1 2

(4:9:68)

(4:9:69)

ðjν1 Þ2 > ðjν0 Þ2 + 4ν ;

ν>0

(4:9:70)

ðjν1 Þ2 > ðjν0 Þ2 + ν2

ν≥0

(4:9:71)

;

The inequalities for both sides are: 2ðν + 1Þ ðν + 3Þ < ðjν, 1 Þ2 < 4 ðν + 1Þ ðν + 2Þ ;

−2 − 1 pffiffiffiffiffiffiffiffiffi 4ðν + 1Þ ν + 2 < ðjν, 1 Þ2 < 2 ðν + 1Þ ðν + 3Þ ; ν > − 1 25=3 ðν + 1Þ½ðν + 2Þðν + 2Þ1=3 < ðjν, 1 Þ2
−1 The lower and upper bounds for the squares of smallest zeros of first derivatives of the Bessel functions are given by ðj′ν1 Þ2 >

4ν ðν + 1Þ ν+2

;

ν>0

pffiffiffiffiffiffiffiffiffi 4ν ðν + 1Þ ν + 2 2 ′ ðj ν1 Þ > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν2 + 8ν + 8

(4:9:78)

; ν>0

ðj′ν1 Þ2 >

25=3 νðν + 1Þ ðν + 2Þ1=2 ðν + 3Þ1=4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν3 + 16ν2 + 38 ν + 21

ðj′ν1 Þ2
0

(4:9:81) ;

ν>0

and the corresponding inequalities for second derivatives are: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j′′ν1 > νðν + 2Þ ; 0 < ν ≤ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j′′ν1 > νðν − 1Þ ; ν > 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ν ðν2 − 1Þ ′′ ; ν>1 j ν1 > 2ν − 1 pffiffiffiffiffiffiffiffiffiffiffi j′′ν1 < ν2 − 1 ; ν > 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi νðν − 1Þ ðν + 2Þ ; ν>1 j′′ν1 < ðν + 1Þ jν0 j′′ν1 > pffiffiffi 2

;

ν>1

(4:9:80)

(4:9:82)

(4:9:83) (4:9:84) (4:9:85) (4:9:86) (4:9:87)

(4:9:88)

202

4 Mathematical Operations with Respect to the Order of the Bessel

j′′ν1 > jν0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8

; ν>1

8 + ðjν0 Þ2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2ν + 1Þ j′′ν1 > jν1 2 ðν + 1Þ

(4:9:89)

; 0 2 νðν − 1Þ ðj′′ν1 Þ2 >

;

4ν ðν − 1Þ ν+2

−1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ν + 13ν2 + 32 ν + 8 2

ðj′′ν1 Þ2 >

;

ν>1

25=3 νðν − 1Þ ½ðν + 1Þðν + 2Þðν + 3Þ1=3 ðν4 + 27 ν3 + 138 ν2 + 134 ν + 24Þ1=3

ðj′′ν1 Þ2
1

(4:9:94)

(4:9:95) ; ν>1

(4:9:96)

There is a large number of inequalities in general case, for any zeros of the Bessel function, for the lower bounds, we have jν, k > ν + k

;

−k ν + k π −

1 2

;

(4:9:97)

;

ν> −

;

jνj ≤

ν>

1 2

;

1 2

1 2

; k = 1, 2, 3, ...

(4:9:98)

k = 1, 2, 3, ...

(4:9:99)

;

k = 1, 2, 3, ...

2ðν + 1Þ ð2 ν + 3Þ 1 ; ν> π 2  ν jν, k > j0, k 1 + ; −k j0, k ð1 − 2 νÞ + 2νπ k jν, k > j0, k + ðj1, k − j0, k Þ ν

;

0 ðj0, k Þ2 + ν2 ; ν > 0

203

(4:9:105) (4:9:106)

and for upper bounds, we have jν, k ≤ βν, k − 

4ν2 − 1 8 βν, k

ν 1 βν, k = π k + − 2 4 jν, k < j0, k +

πν 2

;

0≤ν≤

1 2

;

(4:9:107)



; k = 1, 2, 3, ...

(4:9:108)

ðjνk Þ2 < ðj0, 1 Þ2 + 2 ½π2 k2 − ðj0, 1 Þ2  ν ; jν, k ≤ j0, k +

k = 1, 2, 3, ...

ν

j0, k ½J1 ðj0, k Þ2  ν − 21 jν, k < + πk j0, k ½J1 ðj0, k Þ2

0

(4:9:117)

There is a large number of differential inequalities, their general behaviour can be expressed as d jν, k 1 dν

; ν≥0

(4:9:119)

1 d jν, 1 3ν + 5 > ; ν>1 jν, 1 dν 4ðν + 1Þðν + 4Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d jν, 1 1 1 + 1 + ðjν, 1 Þ2 ; > jν, 1 dν ðjν, 1 Þ2 dðjνk Þ2 >0 ; dν djν, k >1 dν

;

djν, k 1 + > dν jν, k

(4:9:120) ν> −1

ν≥3

(4:9:122)

ν≥0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1+ 2 jν, k

djν, k 2 8ðν + 1Þ2 + > dν jν, k ðjν, k Þ3

(4:9:121)

(4:9:123) ; ν> −1

;

k = 1, 2, 3, ...

(4:9:124)

;

;

k = 1, 2, 3, ...

(4:9:125)

ν> −1

d jν, k 4 8ðν + 1Þðν + 3Þ 32ðν + 1Þ2 ðν + 2Þ2 − + > dν jν, k ðjν, k Þ5 ðjν, k Þ3 ν> −1 ; jν, k

k = 1, 2, 3, ... "  # d jν, k 2 ðν + 1Þ 2 >2 1+ dν jν, k

(4:9:126)

;

ν> −1

; k = 1, 2, 3, ...

(4:9:127)

4.9 Zeros of the Bessel Functions with Respect to the Argument

jν, k

2 !2 3 d jν, k 4ðν + 1Þðν + 3Þ 4ðν + 1Þ ðν + 2Þ 5 + > 2 41 + 2 − dν j2ν, k jν,2 k

ν> −1 ;

205

(4:9:128)

k = 1, 2, 3, ...

and for upper bounds, they are d jν, k jν, k < dν ν+k

;

d jν, k jν, k < dν ν + 21

; k = 1, 2, 3, ... ; ν ≥ 0

k = 1, 2, 3, ... ;

ν> −1

(4:9:129) (4:9:130)

d jν, k π jν, k < ; ν≥0 ; k≥0 dν 2jν, k + ðπ − 2Þν " # d jν, k jν, k 4 8ðν + 1Þ2 + < jν, k + dν ðν + 3Þ jν, k ðjν, k Þ2 ν> −1 ;

k = 1, 2, 3, ... "

d jν, k jν, k 8 16ðν + 1Þ2 32ðν + 1Þ2 ðν + 2Þ2 − + < jν, k + dν ðν + 4Þ jν, k ðjν, k Þ5 ðjν, k Þ3 ν> −1

;

(4:9:131)

(4:9:132) # (4:9:133)

k = 1, 2, 3, ...

The expressions for yν,k zeros of the Bessel function of the second kind are given by yν, 1 ⁓ν + 0.9315768 ν1=3 +

0.230351 0.01198 0.0060 + + ... − ν ν1=3 ν5=3

(4:9:134)

2.022183 0.03572 0.3463 − − 5=3 + ... ν ν1=3 ν

(4:9:135)

4.410233 0.14676 1.6444 − − 5=3 + ... ν ν1=3 ν

(4:9:136)

ν1 yν, 2 ⁓ν + 2.5962685ν1=3 + ν1 yν, 3 ⁓ν + 3.8341592 ν1=3 + ν1 yν, k < y0, k + 1.4470201... ν yν, k < y0, k +

1 y0, k ½Y1 ðy0, k Þ2

;

ν>0

(4:9:137)

; ν>0

(4:9:138)

206

4 Mathematical Operations with Respect to the Order of the Bessel

dyν, k π < dν 2

;

ν>0

;

k = 1, 2, 3, ...

(4:9:139)

d 2 yν, k 0 ;

k = 1, 2, 3, ...

(4:9:140)

and similarly for cν,k zeros, the following inequalities hold cν, k > c0, k + ν ; ν > 0 ; k = 2, 3, 4...   ν 1 1 cν, k ≥ π k + − −α ; 0≤ν≤ ; 2 4 2 cν, k < c0, k +

πν 2

;

ν>0

ν≥

d cν, k cν, k < dν ν + 21

k = 1, 2, 3, ...

d 2 cν, k 0

; ν>0

;

k = 1, 2, 3, ...

; k = 1, 2

  ν 1 −α ; cν, k ≤ π k + − 2 4 ;

(4:9:141)

;

1 2

;

(4:9:142) (4:9:143)

k = 1, 2, 3, ...

k = 1, 2, 3, ...

(4:9:144) (4:9:145)

(4:9:146)

Euler-Rayleigh sums for positive zeros of JνðnÞ ðzÞ, n = 0,1,2,3, . . ., or the Rayleigh function (Lord Rayleigh started investigation of this topic in 1874 [7]) is defined by σn ðνÞ =

∞ X

1

2n k = 1 ðjν , k Þ

; n = 1, 2, 3, ...

(4:9:147)

and it gives the following estimation for the square of the Bessel function zeros 1 ½σn ðνÞ1=n

< ðjν , k Þ2
n – 1, then nth derivative of the Bessel function of the first kind of the order ν has infinitely many zeros which are all real simple except at the origin. If ν > n, then the positive zeros of nth and (n + 1)th derivative are interlacing. For the first and second derivatives of the Bessel function, we have the following EulerRayleigh sums ∞ X

1 ν+2 = 2 4ν ðν + 1Þ ′ k = 1 ðj ν , k Þ ∞ X

1

k = 1 ðj′ν , k Þ ∞ X

1

k = 1 ðj′ν , k Þ

ν2 + 8ν + 8

= 4

½4νðν + 1Þ2 ðν + 2Þ ν3 + 16ν2 + 38 ν + 24

= 6

25 ½ ν ðν + 1Þ3 ðν + 2Þðν + 3Þ

∞ X

1 ν+2 = 2 4ν ðν − 1Þ ′′ k = 1 ðj ν , k Þ ∞ X

1

k = 1 ðj′′ν , k Þ

4

=

13ν3 + 19 ν2 + 26 ν + 8 ½4νðν − 1Þ2 ðν + 1Þ ðν + 2Þ

(4:9:155)

(4:9:156)

(4:9:157)

(4:9:158)

(4:9:159)

It is worthwhile to mention also the general result for zeros of the nth derivative ∞ X

1

ðnÞ 2 k = 1 ðjν , k Þ ∞ X

=

ν+2 4 ðν − n + 1Þðν − n + 2Þ

=

ν+2 Rðν, nÞ 16 ðν − n + 1Þ ðν − n + 2Þ

1

ðnÞ 4 k = 1 ðjν , k Þ

"

# ðν + 2Þ2 ðν + 3Þ ðν + 4Þ − Rðν, nÞ = ðν − n + 1Þ ðν − n + 2Þ ðν − n + 3Þ ðν − n + 4Þ

(4:9:160)

(4:9:161)

208

4 Mathematical Operations with Respect to the Order of the Bessel

In special cases of the Bessel functions, J1/2(z) and J−1/2(z), their positive zeros are known j1/2,k = πk and j−1/2,k = π(k−1/2), and therefore the Rayleigh functions are reduced to   X ∞ 1 1 ςð2nÞ = = 2n ; n = 1, 2, 3, ... (4:9:162) σn 2 n 2 π k = 1 ðj1=2 , k Þ 

σn

 X ∞ 1 1 ð22n − 1Þ ςð2nÞ = − = 2n 2 π2n k = 1 ðj− 1=2 , k Þ

;

n = 1, 2, 3, ...

(4:9:163)

where ζ (z) is the zeta function. In case of the Airy functions, the following Euler-Rayleigh sums are known  Γ 31 1 Aið0Þ  ; Ai′ða′k Þ = 0 =  2 = − ′   Ai ð0Þ 31=3 Γ 32 k = 1 a′k

 7 35 181228 a′k = − ς2=3 1 − + − + ... 48 ς2 288ς4 207360ς6 ∞ X

ς=

(4:9:164)

3 πð4k − 3Þ 8

∞ X

1  3 = 1   k = 1 a ′k

(4:9:165)

  1 2 Γ 3 1   2  4 = 2=3   2 · 3 Γ 32 k = 1 a ′k  ∞ X 2 Γ 31 1  5 = 4=3 2  ′  3 Γ 3 k=1 ak (   1 3 ) ∞ X Γ 3 1 1 1+  6 = 3 4   3 ½Γð2Þ k = 1 a′ ∞ X

k

(4:9:166)

(4:9:167)

(4:9:168)

3

  2 7 Γ 31 1     7 =  ′  32=3 · 15 Γ 2 2 k=1 ak 3

(4:9:169)

   1  4 11 Γ 31 Γ 3 1  +   4  8 = 1=3 2 4=3 · 36 Γ 3   3 · 8 Γ 32 3 k = 1 a ′k

(4:9:170)

∞ X

∞ X

Contrary to significant interest devoted to zeros of the Bessel functions of the first and the second kind Jν(z) and Yν(z), their linear combinations or cross products,

4.9 Zeros of the Bessel Functions with Respect to the Argument

209

only limited attention has been directed to zeros of the Struve functions hν,k, and even less to zeros of the Lommel functions, σν,k.(μ) In the case of the Struve functions Hν ðhν, k Þ = 0

(4:9:171)

the positive zeros are all simple and they satisfy the following chains of inequalities jν, k < π k < hν, k < jν, k + 1 < π ðk + 1Þ < hν, k + 1 < jν, k + 2 < ... 1 2

jνj
0

(4:10:15)

A quite different approach in treating the Bessel functions of nearly equal argument and order was presented by Paris [146]. He derived a number of new inequalities, most of them for the 0 < x ≤ 1 interval νx Jν ðν xÞ ν+2

Jν + 1 ðνxÞ
0 ∂ν

215

(4:10:22) (4:10:23)

and

 ∂ Jν ðνxÞ ≤0 ∂ν Jν ðνÞ

 ∂ J ′ ðνÞ ≥0 ν1=3 ν ∂ν Jν ðνÞ 

  ∂ ∂ Jν ðν xÞ =Jν ðν xÞ ≥ 0 ∂x ∂ν     1 ∂ Jν ðνxÞ 1 ∂ Jν ðνÞ ≤ Jν ðνxÞ ∂x Jν ðνÞ ∂ν     ∂2 Jν ðν xÞ ∂ Jν ðν xÞ ∂ Jν ðνxÞ ≥ 0 Jν ðν xÞ − ∂ν ∂x ∂ν ∂x

(4:10:22)

(4:10:25) (4:10:26) (4:10:27)

(4:10:28)

The second group of formulas is related to the Nicolson integral (4.8.42) ½Jν ðxÞ2 + ½Yν ðxÞ2 =

8 π2

∞ ð

K0 ð2 x sinh ξÞ coshð2 ν ξÞ dξ

(4:10:29)

0

n

o 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi > ½Jν ðxÞ2 + ½Yν ðxÞ2 < πx π x 2 − ν2 ∞ ð ∂Yν ðxÞ ∂Jν ðxÞ 4 e − 2 νξ K0 ð2x sinh ξÞ dξ Jν ðxÞ − Yν ðxÞ =− ∂ν ∂ν π

(4:10:30)

(4:10:31)

0

Jν ðxÞ

∂Yν ðxÞ ∂Jν ðxÞ − Yν ðxÞ >0 ∂ν ∂ν

∂Y ′ν ðxÞ ∂J ′ν ðxÞ − Y ′ν ðxÞ ∂ν ∂ν ∞ ð 4 e − 2 νξ ½x2 coshð2 ξÞ − ν2  K0 ð2 x sinh ξÞ dξ =− π x2

(4:10:32)

J ′ν ðxÞ

0

(4:10:33)

216

4 Mathematical Operations with Respect to the Order of the Bessel

4.11 The Asymptotic Limit of the Bessel Function ν Jν(νx) Expressed as the Shifted Dirac Function δ(x – 1). Evaluation of Integral Representations of Elementary Functions, Special Functions, Mathematical Constants, Integral Transforms, Asymptotic Relations, Integrals and Limits of Functional Series A sudden excitation of physical systems is expressed mathematically by the impulse functions. These functions were introduced to mathematics by Fourier, Cauchy, Poisson and Hermite. However, their use was significantly extended when Helmholtz, Kirchhoff, Kelvin and Heaviside started to apply them in solutions of various practical problems in electricity, fluid mechanics and heat transfer [82, 160–162]. The recognition of impulse functions as a powerful tool in many branches of applied mathematics, physics and engineering was firmly established when Dirac introduced them to quantum mechanics in 1926. From that time, the impulse function is named after him, δ(x) – Dirac delta function. The impulse functions are not ordinary functions in mathematical sense, but functions defined by giving a rule for integrating their product with a continuous function f(x). These functions are approximated by the sequence of functions converging to them (i.e. delta sequences). δðx − 1Þ = lim δn ðx − 1Þ

(4:11:1)

n!∞

The above sequence is written in the form for the shifted delta function δ(x – 1), which will be used in this section. A number of delta sequences are known in mathematics, three “classical” sequences are presented below   sin½νðx − 1Þ (4:11:2) δðx − 1Þ = lim ν!∞ π ðx − 1Þ ! ! 1 ν 1 ε δðx − 1Þ = lim = lim (4:11:3) π ν ! ∞ 1 + ν2 ðx − 1Þ2 π ε!0 ε2 + ðx − 1Þ2 1 νe − jνðx − 1Þj  δðx − 1Þ = lim 2 ν!∞ Γ 1 + n1

n

! ;

n = 1, 2, 3, ...

(4:11:4)

The first sequence was proposed by Fourier, the second sequence by Cauchy and the last by Kelvin, when for n = 2, the sequence is called the “heat source.” Since our main interest is directed to the Bessel and related functions, the shifted Dirac function δ(x – 1) can be represented by the orthogonality identities for the Bessel functions of the first and second kind and the Struve function of the first kind

4.11 The Asymptotic Limit of the Bessel Function

217

∞ ð

δðx − 1Þ =

ðxt ÞJν ðtÞJν ðx tÞ dt 0

x>0

;

(4:11:5)

Re ν > − 1

and ∞ ð

ðxtÞ Yν ðtÞHν ðx tÞ dt

δðx − 1Þ = 0

x>0

;

(4:11:6)

Re ν > − 1

However, the delta sequence, which is directly related to the order of Bessel function and therefore is of particular interest to us, was derived in 1969 by Lamborn [83] δðx − 1Þ = lim ½νJν ðνx Þ ν!∞

(4:11:7)

Using this shifted Dirac delta function, the present author over the 1999–2008 period, in a series of papers, was able to derive many new mathematical relations. They included representations of the mathematical constants, the elementary and special functions, the asymptotic and functional relations, infinite integrals, series and limits [84,85,29]. Following these investigations, Laforgia and Natalini [163] obtained additional limiting relations, mainly for the Bessel functions. The equivalent form of the Lamborn expression in terms of the Bessel-Clifford function is h i 1 lim νν + 1 xν − 1=2 Cν ðν2 x Þ 4 ν!∞ pffiffiffi − ν=2 Jν ð2 xÞ Cν ðx Þ = x δð4x − 1Þ =

(4:11:8)

It is also possible to express the delta function by using the infinite integral with integrand having product of the Airy functions ∞ ð

Aiðt − xÞ Aiðt − aÞ dt

δðx − aÞ =

(4:11:9)

− ∞

The shifted Dirac delta function δ(x – a) is defined by ( 0 for all x ≠ a δðx − aÞ = ∞ for x = a

(4:11:10)

and its most important property is the so-called sifting property or the sampling property

218

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð

δðx − aÞf ðxÞdx = f ðaÞ

(4:11:11)

−∞

The following formula represents derivatives of the delta function ∞ ð

δðnÞ ðx − aÞf ðxÞdx = ð − 1Þn f ðnÞ ðaÞ

(4:11:12)

−∞

n = 0, 1, 2, 3 . . . where it is assumed that f(x) is continuously differentiable in the neighbourhood of point a, – ∞ < a < ∞. The sifting property in (4.11.11) is independent of the actual values of the limits of integration, it depends only on the behaviour of the integrand near the point a. Thus, the delta function and its derivatives are defined by a rule, by integration of the products appearing in (4.11.11) and (4.11.12). If the Lamborn delta sequence in (4.11.7) is multiplied by a function f(tx) and integrated from zero to infinity with respect to variable x we have " ∞ # ∞ ð ð f ðtxÞδðx − 1Þ dx = lim ν

f ðtÞ =

f ðtxÞJν ðνxÞ dx

ν!∞

0

(4:11:13)

0

It is assumed here and in other places that inversion of mathematical operations is permissible. Thus, it follows from (4.11.13) that function f(t) is represented by the asymptotic limit of the infinite integral of product of f(tx) and the Bessel function Jν(νx). If the right-hand integral in (4.11.13) is evaluated in the closed form then the limit can be regarded as the generalization of the l’Hôpital’s rule. f ðtÞ = lim ½ν Φðt, νÞ ν!∞ ∞ ð

f ðtxÞJν ðνxÞ dx

Φðt, νÞ =

(4:11:14)

0

Since many infinite integrals having the integrand Bessel functions and various f(tx) functions are tabulated, it is possible to present many functions f(t) as the limit given in (4.11.14). A number of examples, mainly with the Bessel functions, will illustrate the applied procedure. Let us start with the Bessel function itself, f(tx)= Jμ(tx) ∞ ð

Jμ ðt xÞJν ðν xÞ dx

Φðt, ν, μÞ = 0

(4:11:15)

4.11 The Asymptotic Limit of the Bessel Function

219

but this is the Weber-Schafheitlin type integral [7] ∞ ð

Jν ðν xÞJμ ðt xÞ dx   μ+ν+1 μ   t Γ μ+ν+1 μ−ν+1 t2 2   = , ;μ + 1; 2 2 F1 ν − μ + 1 μ+1 ν 2 2 ν Γðμ + 1Þ Γ 2 Re μ > − 1 ; 0 < t < ν 0

(4:11:16)

Thus, using (4.11.14) and (4.11.16), the Bessel function is represented as the limit of the hypergeometric function  9 8  μ+ν+1 > > > > Γ = < tμ 2   Iðt, μ, νÞ lim Jμ ðtÞ = > ν−μ+1 μ Γðμ + 1Þ ν ! ∞> > > ; :Γ ν 2 (4:11:17)   μ+ν+1 μ−ν+1 t2 Iðt, μ, νÞ = 2 F1 , ;μ + 1; 2 ν 2 2 Re μ > − 1 ;

0 0 ;

Re ν > − 1 ;

(4:11:28)

3 Re μ < 2

it is possible to obtain the limit representation of the product of the modified Bessel functions  pffiffiffiffiffiffiffiffiffiffiffiffiffi μ

 μ     t + a2 + t 2 νa aν aν pffiffiffiffiffiffiffiffiffiffiffiffiffi Iðν − μÞ=2 (4:11:29) Kðν + μÞ=2 = lim ν!∞ t 2t 2t a2 + t 2 Denoting λ = ν /2, τ = a/t, and ρ = μ/2, the limit in (4.11.29) takes the form

222

4 Mathematical Operations with Respect to the Order of the Bessel



pffiffiffiffiffiffiffiffiffiffiffi 2ρ 1 + 1 + τ2 pffiffiffiffiffiffiffiffiffiffiffi lim λ Iλ − ρ ðλτÞKλ + ρ ðλτÞ = λ!∞ 2 τ2 ρ 1 + τ2 



Re τ > 0

; Re ρ
0

(4:11:31)

The left side expression in (4.11.30) is the first term in the uniform asymptotic expansion of the product of modified Bessel functions for large orders [9] eλ η Iλ ðλτÞ⁓ pffiffiffiffiffiffiffiffiffi f1 + . . . g 2 π λð1 + τ2 Þ1=4 rffiffiffiffiffi π e− λη Kλ ðλτÞ⁓ f1 + . . . g 2 λ ð1 + τ2 Þ1=4   pffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi η = 1 + τ2 + ln 1 + 1 + τ2

(4:11:32)

By introducing τ = 1 into (4.11.32), we have for large values of λ the expected result ( ) 1 1 ð4 λ2 − 1Þ 3 ð4λ2 − 1Þð4 λ2 − 9Þ λ Iλ ðλτÞ Kλ ðλτÞ⁓ pffiffiffiffiffiffiffiffiffiffiffi 1 − + − ... 2 ð2λτÞ2 8 ð2λτÞ4 2 1 + τ2 (4:11:33) λ!∞ Using the following integral [13] ∞ ð

e− t

2 x2

Jν ðνxÞ dx =

pffiffiffi  2 ν π − ν2 =8 t2 Iν=2 e 8t 2t

(4:11:34)

0

it is possible, after few steps, to obtain the limit representation of the modified Bessel function of the first kind ( !) rffiffiffi λ2 τ −τ − λ2 =2 τ lim λe Iλ (4:11:35) = e λ!∞ 2τ π and using

4.11 The Asymptotic Limit of the Bessel Function

∞ ð

e

− 1=t x

dx Jν ðν xÞ = 2 Jν x

rffiffiffiffiffi! rffiffiffiffiffi! 2ν 2ν Kν t t

;

ν, t > 0

223

(4:11:36)

0

the limit of following product of the Bessel functions is  pffiffiffiffiffiffi pffiffiffiffiffiffi  e − τ=2 lim νJν ð ντÞ Kν ð ν τÞ = ν!∞ 2

; τ>0

(4:11:37)

The next examples will include the limits of elementary and special functions based on using (4.11.14). From the Laplace transform of the Bessel function [37] ∞ ð

0

νν e − t x Jν ðνxÞ dx = pffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffiiν ν2 + t2 t + ν2 + t2

;

t>0

the asymptotic limit for the exponential function is 9 8 > > = < ν+1 ν ; t>0 e − t = lim pffiffiffiffiffiffiffiffiffiffiffiffih pffiffiffiffiffiffiffiffiffiffiffiffiiν ν ! ∞> ; : ν2 + t2 t + ν2 + t2 >

(4:11:38)

(4:11:39)

From the Bessel function integrals [13]    sin νsin − 1 νt pffiffiffiffiffiffiffiffiffiffiffiffi sinðtxÞ Jν ðν xÞ dx = ; ν>t>0 ν2 − t 2 0 ∞    ð cos νsin − 1 νt pffiffiffiffiffiffiffiffiffiffiffiffi cosðtxÞ Jν ðν xÞ dx = ; ν>t>0 ν2 − t 2 ∞ ð

(4:11:40)

0

the asymptotic limits for the trigonometric functions are: (    ) ν sin νsin − 1 νt pffiffiffiffiffiffiffiffiffiffiffiffi lim = sin t ; ν > t > 0 ν!∞ ν2 − t 2 (   ) ν cos νsin − 1 νt pffiffiffiffiffiffiffiffiffiffiffiffi = cos t ; ν > t > 0 lim ν!∞ ν2 − t 2 and by dividing these limits, we have

   t lim tan νsin − 1 = tan t ; ν > t > 0 ν!∞ ν

   −1 t = cot t ; ν > t > 0 lim cot νsin ν!∞ ν

(4:11:41)

(4:11:42)

224

4 Mathematical Operations with Respect to the Order of the Bessel

In the following two examples the limits will include special functions, but not the Bessel functions. Using the following integral ∞    ð Γ λ + ν νν λ+ν ν2 2 2 xλ − 1 e − t x Jν ðνxÞ dx = ν + 1 2 F ;ν + 1; − 1 1 4t2 2 2 Γðν + 1Þ tν + λ (4:11:43) 0

Reðλ + νÞ > 0 ;

t>0

from (4.11.14) the limit of the confluent hypergeometric function is (   ) Γ λ +2 ν  ν ν + 1 λ+ν ν2 2 lim = tλ − 1 e − t ;ν + 1; − 2 1 F1 ν ! ∞ Γðν + 1Þ 2 t 4t 2 Re λ > 0

(4:11:44)

; t>0

In the second example the integral includes the Whittaker functions 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ 1 + ðt xÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jν ðνxÞ dx λ 1 + ðt xÞ2 x 0   ν ν Γ ν − 2λ + 1 tλ M − λ=2 , ν=2 = Wλ=2 , ν=2 ν Γðν + 1Þ t t ∞ ð

1+

Re ðν − λÞ > − 1

(4:11:45)

; t>0

and therefore from (4.11.4) it follows that (  ) ν  ν Γ ν − 2λ + 1 tλ M − λ=2 , ν=2 lim Wλ=2 , ν=2 ν!∞ Γðν + 1Þ t t h

pffiffiffiffiffiffiffiffiffiffiiλ 1 + 1 + t2 pffiffiffiffiffiffiffiffiffiffi = tλ 1 + t2

(4:11:46)

The Lamborn formula for the shifted Dirac delta function given in (4.11.7) can be explored to obtain additional representations of functions by considering its derivatives with respect to the argument x. The first derivative of the shifted Dirac delta function is δðx − 1Þ = lim fν Jν ðνxÞg ν! ∞

δ′ðx − 1Þ =

  1 lim ν2 ½Jν − 1 ðνxÞ − Jν + 1 ðνxÞ ν !∞ 2

(4:11:47)

4.11 The Asymptotic Limit of the Bessel Function

225

but δðnÞ ðxÞ = ð − 1Þn n!

δðxÞ xn

(4:11:48)

and in similar way as in (4.11.13), we have f ′ðtÞ =

∞ ð

δ′ðx − 1Þ f ðt xÞ dx

0

1 =− lim 2t ν ! ∞

∞ ð

 2  ν ½Jν − 1 ðνxÞ − Jν + 1 ðνxÞ f ðt xÞ dx

(4:11:49)

0

Higher derivatives of function, f(n)(t), can be derived in the same manner by using (4.11.48) and by considering that it is known [11]. n ν n X dn Jν ðν xÞ ð − 1Þk = n! Jν − n + 2 k ðνxÞ dxn 2 k = 0 ðn − kÞ! k!

(4:11:50)

An application of the first derivative of the shifted Dirac delta function can be illustrated by using (4.11.34) ∞ ð

e

− t 2 x2

pffiffiffi  2 ν π − ν2 =8 t2 Jν ðν xÞ dx = Iν=2 e 8t 2t

0

f ðtÞ = e − t

(4:11:51)

2

f ′ðtÞ = − 2t e − t

2

and therefore from (4.11.48) we have

lim ν e 2

− ν2 =8 t2

ν!∞

  2  2  2 ν ν 8 t3 e − t − Iðν + 1Þ=2 = pffiffiffi Iðν − 1Þ=2 8t 8t π

(4:11:52)

So far, in order to derive asymptotic limit expressions, the same function f(tx) appeared on both sides of equations (4.11.13) and (4.11.49). This can be changed if integral transforms are combined with the Lamborn formula. If integral transformation of a function f(t) is defined by ∞ ð

Kðs, tÞf ðtÞ dx = TðsÞ

Tff ðtÞg = 0

and the function is f(t) = Jν(νt), then we have

(4:11:53)

226

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð

δðt − 1ÞKðs, tÞ dt 0

" ∞ # ð = lim ν Kðs, tÞ Jν ðν tÞ dt = Kðs;1Þ

(4:11:54)

ν!∞

0

where K(s,t) denotes the kernel of integral transformation. Most of integral transformations satisfy the following similarity rule Tff ða tÞg =

1 s T a a

; a>0

(4:11:55)

and therefore by comparing (4.11.54) with (4.11.55), the equivalent forms of asymptotic limits are [85] " ∞ # " ∞ # ð ð s s dx (4:11:56) TðsÞ = lim ν Jν ðνξÞ TðsξÞ dξ = lim ν Jν ðν xÞ T ν!∞ ν !∞ x x 0

0

An application of the asymptotic limit in (4.11.54) can be demonstrated by using the Laplace transform of f(t) = Jν(νt) from (4.11.38) Kðs, tÞ = e − s t

; ∞ ð

LfJν ðν tÞ, sg = 0

s>0

νν e − s t Jν ðν tÞ dt = pffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffiffiiν ν2 + s2 s + ν2 + s2

(4:11:57)

Re ν > 0 and therefore, 9 8 > > = < νν + 1 lim pffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffiffiiν = e − s ν ! ∞> ; : ν2 + s2 s + ν2 + s2 >

;

s>0

(4:11:58)

For the Stieltjes transform of Jν(νt), we have 1 ; s>0 s+t ∞ ð Jν ðνtÞ π ½Jν ðνsÞ − Jν ðνsÞ dt = SfJν ðν tÞg = s+t sinðπ νÞ

Kðs, tÞ =

0

Re ν > 0

(4:11:59)

4.11 The Asymptotic Limit of the Bessel Function

which gives immediately from (4.11.54)

 ν ½Jν ðνsÞ − Jν ðν sÞ 1 = lim ; ν !∞ sinðπ νÞ π ðs + 1Þ

s>0

227

(4:11:60)

The Anger function is represented by sinðπ νÞ Jν ðν sÞ − Jν ðνsÞ = π

∞ ð

e − ν τ − ν s sinh τ dτ

(4:11:61)

0

and therefore, the limit of this integral is derived as a by-product ( ∞ ) ð 1 − ν τ − ν s sinh τ lim ν e dτ = ; s>0 ν!∞ s+1

(4:11:62)

0

The limit in (11.4.54) contains only the kernel of integral transformation K(s,t), but if also the function f(ξ;λ) where λ is a parameter, is introduced, then usefulness of derived expression is considerably enlarged. Multiplication of (4.11.54) by f(ξ;λ⁏) and integration from zero to infinity with respect to variable ξ gives ( ∞ ) ð∞ ð lim

ν!∞

ν

Kðξ, xÞ Jν ðνxÞ f ðξ, λÞ dx dξ

0 0 ( ∞ ð

ν

= lim

ν!∞

"∞ ð f ðξ, λÞ

# Kðξ, xÞ Jν ðνxÞ dx

0

0

0

0

) dξ

(4:11:63)

( ∞ "∞ # ) ð ð Kðξ, xÞ f ðξ, λÞ dξ dx = lim ν Jν ðνxÞ ν!∞

or in the compact form ∞ ð

lim ½ν

f ðξ, λÞ TfJν ðνxÞ, ξg dξ

ν!∞

0

(4:11:64)

∞ ð

= lim ½ν

Jν ðν xÞTff ðξ, λÞ, xg dx = Tð1, λÞ

ν!∞

0

For example, for the Laplace transformation we have

228

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð

νν e − ξ x Jν ðνξÞ dξ = qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν 2 + ξ 2 ½ξ + ν ν2 + ξ 2  0 9 8 ∞ > > ð = < f ðξ, λÞ lim νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν dξ ν !∞ > > ; : ν2 + ξ 2 ½ξ + ν2 + ξ 2  0 ( ∞ ) ð

Lff ðξ, λÞ, xg =

= lim

ν!∞

ν

(4:11:65)

Jν ðνxÞ Lðx, λÞ dx = Lð1, λÞ 0

By suitable choice of the original function transform pair, the above expression permits to represent mathematical constants and special functions as limits of integrals with the integrands being elementary functions from the Laplace transform

   sin ξ 1 = tan − 1 (4:11:66) L ξ x it follows from (4.11.65) that π number is expressed by the following limit 9 8 > > ∞ > > ð = < sin ξ π ν+1 = tan − 1 ð1Þ = lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > 4 > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ξ

(4:11:67)

The Euler constant γ = 0·577215·is represented by starting from the Laplace transform 1 Lfln ξg = − ðγ + ln xÞ x and therefore, 9 8 > > ∞ > > ð = < ln ξ ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ = − γ lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q ν !∞ > > > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 The Bernoulli numbers can be evaluated using ( )   ξμ−1 x+1 L ; μ>1 = 21 − μ ΓðμÞ ζ μ, sinh ξ 2

(4:11:68)

(4:11:69)

(4:11:70)

where the generalized zeta function ζ(z,α) becomes ζ(z,1) = ζ(z) and therefore,

4.11 The Asymptotic Limit of the Bessel Function

9 8 > > ∞ > > ð = < μ−1 ξ ν+1 lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 sinh ξ

229

(4:11:71)

= 21 − μ ΓðμÞ ςðμÞ but ζ ð2 nÞ =

ð2 πÞ2 n jB2 n j 2 ð2nÞ!

;

n = 1, 2, 3, ...

(4:11:72)

and therefore, 9 8 > > > > ∞ > > ð = π2n < 2n − 1 ξ ν+1 = lim ν dξ jB j   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi λ ν! ∞> > 2n 2n > > 2 2 > > 2 2 0 ; : ν +ξ ξ + ν +ξ sinh ξ

(4:11:73)

Limit representations of special functions are given in Appendix B, but those derived for the Bessel and related functions are presented below. Most of them are determined by applying the Laplace transforms, but the analogs of (4.11.65) of the Fourier cosine and sine and the Stieltjes transformations are also used 2 3 h  i ∞ ð cos νsin − 1 ξ f ðξ, λÞ ν 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ 5 = Fc ð1, λÞ lim 4ν ν!∞ 2 ν2 − ξ 0 2 3 h  i ∞ ð sin νsin − 1 ξ f ðξ, λÞ ν 6 7 (4:11:74) qffiffiffiffiffiffiffiffiffiffiffiffiffi lim 4ν dξ 5 = Fs ð1, λÞ ν!∞ 2 2 ν −ξ 0 ( ∞ " # ) ∞ ð ð lim

ν!∞

ν

e − ν η − ν ξ sinh η dη dξ

f ðξ, λÞ 0

= Sð1, λÞ

0

The asymptotic limits of the Bessel functions of the first kind are: Jμ ðtÞ

9 8 h  i > ðt cos ν sin − 1 ξ ðt2 − ξ 2 Þμ − 1=2 > = < ν 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi = μ − 1 pffiffiffi  lim ν dξ 1 μ ν!∞ > > 2 πΓ μ+ 2 t ; : ν2 − ξ 2 0

Re μ > −

1 2

(4:11:75)

230

4 Mathematical Operations with Respect to the Order of the Bessel

and Y − ðμ + 1Þ ðtÞ

=

1   p ffiffiffi 1 μ+1 μ − 1 t 2 πΓ μ+ 2 −

   9 8 2 −1 ξ > > 2 μ − 1=2 ∞ > > ð ξ ðξ sin νsin − t Þ = < ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi lim ν dξ ν!∞ > > > > ν2 − ξ 2 ; : t

1 < Re μ < 0 2 (4:11:76)

In the case of modified Bessel functions, we have Iμ ðtÞ =

8 > >
> : ð2tÞ π Γ μ+ 2 μ> −

2ðt

0

9 > > =

2 μ − 1=2

ð2t ξ − ξ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

(4:11:77)

1 2

9 8 > > ∞ > > pffiffiffiffi − t ð μ − 1=2 = < 2 ðξ + 2 t ξÞ πe ν + 1  lim ν Kμ ðtÞ =   qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > 1 > > ; : ð2tÞμ Γ μ+ ν2 + ξ 2 ξ + ν2 + ξ 2 0 2 μ> −

(4:11:78)

1 2

Similarly, the Struve functions Hμ(t) and Lμ(t) are represented by Hμ ðtÞ = Yμ ðtÞ +

8 > >
> μ − 1 : t 2 π Γ μ+ 2

μ> −

1 2

,

∞ ð

0

2 μ − 1=2

9 > > =

ðt2 + ξ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

t>0 (4:11:79)

and

231

4.11 The Asymptotic Limit of the Bessel Function

Lμ ðtÞ = Iμ ðtÞ −

2μ − 1

8 > >
> =

2 μ − 1=2

1 ðt2 − ξ Þ lim νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ pffiffiffiffi  1 μ ν ! ∞> > π Γ μ+ 2 t > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

Re μ > −

1 2

,

t>0 (4:11:80)

The asymptotic limits for the Anger function Jμ(t) and Weber function Eμ(t) are Jμ ðtÞ = Jμ ðtÞ

−

8 > > >
πμ > > :

∞ ð

0

 qffiffiffiffiffiffiffiffiffiffiffiffi − μ  9 > 2 > −μ 2 > ξ + t +ξ −t = dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi λ > > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

(4:11:81)

and Eμ ðtÞ = − Yμ ðtÞ + t lim

8 > >
>

2

:

νν + 1

0

ξ Iðξ, t, μÞ = 4 − @ + t

∞ ð

0

9 > > =

Iðξ, t, μÞ qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ξ2 ; 1 + t 2 ν2 + ξ 2 ξ + ν 2 + ξ 2

sffiffiffiffiffiffiffiffiffiffiffiffi1μ sffiffiffiffiffiffiffiffiffiffiffiffi0sffiffiffiffiffiffiffiffiffiffiffiffi ξ 2A ξ2 ξ2 + cosðπ μÞ 1 + 2 @ 1 + 2 − 1+ 2 t t t

The Lommel functions limits are presented here in two cases only 9 8 h  i > > ∞ ξ − 1 > > ð = < sinh μsinh t 1 ν+1 S0, μ ðtÞ = dξ   lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > > μt ν ! ∞> > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

S1, μ ðtÞ =

8 > >
> :

∞ ð

0

9 h  i > −1 ξ > = cosh μsinh t dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(4:11:82) 1μ 3 ξA 5 t

(4:11:83)

(4:11:84)

If the shifted Dirac delta function is expressed by the orthogonality identity of the Bessel functions (4.11.5)

232

4 Mathematical Operations with Respect to the Order of the Bessel

∞ ð

δðx − 1Þ =

x u Jν ðuÞJν ðx uÞ du

(4:11:85)

0

then by multiplication of both sides of (4.11.85) by function f(tx) and performing integration from zero to infinity with respect to variable x gives # ∞ ∞ ð "∞ ð ð δðx − 1Þ f ðt xÞ dx =

f ðtÞ =

x uJν ðuÞJν ðx uÞ du f ðt xÞ dx

0

0

(4:11:86)

0

which can be rearranged to gðtÞ = t1=2 f ðtÞ

(∞ ð

∞ ð

=

u

1=2

Jν ðuÞ

ðx uÞ

0

1=2

h

Jν ðx uÞ ðt xÞ

1=2

i

f ðt xÞ dx

)

(4:11:87) du

0

The inner integral can be recognized as the Hankel transform of order ν of the function g(ξ) ∞ ð

ðs ξÞ1=2 Jν ðsξÞ gðξÞ dξ = Gðs, νÞ

GfgðξÞ, νg =

(4:11:88)

0

s>0

;

1 Re ν > − 2

Taking into account the operational rule of the Hankel transformation GfgðaξÞ, νg =

1 s G ,ν a a

;

a>0

(4:11:89)

u , ν du t

(4:11:90)

we have ∞ ð

t

3=2

f ðtÞ = t gðtÞ =

u1=2 Jν ðuÞ G 0

which can be treated as some kind of inversion formula The corresponding asymptotic limit of (4.11.90) is

233

4.11 The Asymptotic Limit of the Bessel Function

t

1=2

" ∞ # ð 1=2 f ðtÞ = lim ν Jν ðν xÞ ½ðt xÞ f ðt xÞ dx ν!∞

0

∞ ð ν 1=2 G , ν = ðν tÞ Jν ðνxÞ ½ðt xÞ1=2 f ðt xÞ dx t 0

n

f ðtÞ = lim ν1=2 G ν!∞

(4:11:91)

ν o ,ν t

If ν = ± 1/2, the Hankel transforms are reduced to the Fourier sine and cosine transforms, and (4.11.90) becomes t3=2 f ðtÞ = t gðtÞ =

∞ ð

2 π

sin u Fs 0 ∞ ð

2 t3=2 f ðtÞ = t gðtÞ = π

sin u Fc

u du t u t

(4:11:91) du

0

If the integral transform with the Struve function as the kernel is considered ∞ ð

ðs ξÞ1=2 Hν ðsξÞ gðξÞdξ = Hðs, νÞ

Hν fgðξÞg =

(4:11:92)

0

the analogs of (4.11.90) and (4.11.91) have similar forms ∞ ð

t3=2 f ðtÞ = t gðtÞ =

u1=2 Yν ðuÞH

u , ν du t

0

n ν o t f ðtÞ = lim ν1=2 H , ν ν!∞ t

(4:11:93)

Since tables of the Hankel and Struve transforms, (G (s,ν) and H(s,ν) – transforms of order ν of functions g(t) = t1/2f(t)) are available in the literature, the above formulas are a source of many rather complicated finite and infinite integrals of Bessel functions and corresponding limits of elementary and special functions. In order to illustrate the asymptotic limit of elementary function, let us start with the following Hankel function transform pair gðtÞ = tn + 1=2 e − α t

;

n = 0, 1, 2, 3, ... ; Re α > 0 , Re ν > 0 9 8 > > = < dn + 1 1 Gfν, sg = ð − 1Þn + 1 sν + 1=2 n + 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffiffiiν > dα ; : α2 + s2 α + α2 + s2 >

and using (4.11.91) we have

(4:11:94)

234

4 Mathematical Operations with Respect to the Order of the Bessel

8 2 39 > > = < dn + 1 ν+1 ν 6 7 lim q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5 ν n+1 ν ! ∞> > :dα ðα tÞ2 + ν2 ½α t + ðα tÞ2 + ν2  ;

(4:11:95)

= ð − 1Þn + 1 tn + 1 e − α t n = 0, 1, 2, 3, ... ;

Re α > 0

In the case of special function, starting from 2

gðtÞ = tμ − 1=2 e − αt . ; Re α > 0 , Re ðμ + νÞ > 0  2 Γ μ + 2ν + 1 e − s =8 α  Gfν, sg = s2 Γðν + 1Þ αμ=2 s1=2 Mμ=2, ν=2 4α the asymptotic limit for the Whittaker function is (  2  2 ) 2 Γ μ + 2ν + 1 e − ν =8 α t ν 2 Mμ=2, ν=2 lim = αμ t2μ e − αt . ν!∞ Γðν + 1Þ 4αt2

(4:11:96)

(4:11:97)

Lommel functions can be treated by using the following Hankel transform: gðtÞ =

tμ − 1=2 . α+t

; Re μ
− 1

Gfν, sg  8  9 < Γ μ + ν + 1 = 2 Γ μ + 2ν + 2 p ffiffi ffi ν − μ S − μ − 1 , ν ðα sÞ = ð2αÞμ s 1 − μ2 + ν S − μ , ν ðα sÞ − :Γ ; Γ 2 2 and therefore,  8  9 < Γ μ + ν + 1 α ν 2Γ μ + 2ν + 2 α ν =  S− μ−1, ν lim ν −  2 S− μ, ν ν ! ∞ :Γ 1 − μ + ν t t ; Γ ν −2 μ 2 tμ + 1 = ð2 αÞμ ðα + 1Þ which for μ = 0 reduces to

(4:11:98)

(4:11:99)

4.11 The Asymptotic Limit of the Bessel Function

n lim

ν!∞

S0 , ν

α ν α ν o t − ν S− 1, ν = t t α+1

235

(4:11:100)

The procedure presented here is also efficient in evaluation of integrals. For example, from the Hankel transform function pair gðtÞ =

ξ ν + 1=2 ðα2 + ξ 2 Þ

Gðs, νÞ =

μ

21 − μ αν − μ + 1 sμ − 1=2 Kν − μ + 1 ðαsÞ ΓðμÞ Re ν > − 1 ,

(4:11:101)

Re ðμ − νÞ > 0

by using (4.11.90) we have the following infinite integral of the Bessel functions: ∞ ð

0

uμ Jν ðuÞ Kμ − ν + 1

  αu 2μ − 1 ΓðμÞ αμ − λ − 1 ξ μ + ν + 1 du = μ ξ ðα2 + ξ 2 Þ

(4:11:102)

By changing variables, this integral can be written in a much simpler form ∞ ð

0

xν − μ + 1 Jν ðαxÞKμ ðβ xÞ dx =

2ν − μ Γðν − μ + 1Þ αν βμ ðα2 + β2 Þ Re ν > − 1

ν−μ+1

(4:11:103) ,

3 Re ðν − 2μÞ > − 2

For ν = ± 1/2, the above integral is reduced to integrals with trigonometric functions ∞ pffiffiffi 3 ð πΓ 2 −ν α x1 − ν sinðαxÞ Kν ðβxÞ dx = 3=2 − ν ð2 βÞν ðα2 + β2 Þ 0 (4:9:104) ∞ pffiffiffi 1 ð − ν π Γ 2 x − ν cosðαxÞ Kν ðβ xÞ dx = 1=2 − ν ν + 1 βν ðα2 + β2 Þ 2 0 If the Hankel transform includes the Heaviside step function u(x) or the shifted step function u(x-a)     αt αt Jν=2 − 1=4 , Re ν > − 1 gðtÞ = t1=2 Jν=2 − 1=4 2 2 (4:11:105) 1 Gðs, νÞ = α t pffiffiffiffiffiffiffiffiffiffiffiffiffi ½uðsÞ − uðs − αÞ ðα − sÞ π 2 then the Bessel function integral becomes the finite integral

236

4 Mathematical Operations with Respect to the Order of the Bessel

ðα 0

Jν ðt xÞ pffiffiffiffiffiffiffiffiffiffi dx = π α−x

rffiffiffi     α αt αt Jν=2 + 1=4 Jν=2 − 1=4 2 2 2

,

Re ν > − 1

(4:11:106)

The above integral for ν = ± 1/2, reduces to the trigonometric integrals ðα 0

ðα 0

    sinðt xÞ αt αt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx = π sin J0 2 2 xðα − xÞ (4:11:107)

    cosðt xÞ αt αt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx = π cos J0 2 2 xðα − xÞ

If two shifted Heaviside step functions appear in the Hankel function transform pair gðtÞ = t1=2 − ν Jν ða tÞJν ðb tÞ s1=2 − ν f½s2 − ða − bÞ2 ½ða + bÞ2 − s2 g pffiffiffi  Gðs, λÞ = 23 ν − 1 π Γ ν + 21 ðabÞν Iðs, a, bÞ = fu½s − ja − bj − u½s − ða + bÞg

ν − 1=2

Iðs, a, bÞ

; Re ν > −

(4:11:108)

1 2

the finite integral has limits of integration which are not at the origin as in (4.11.106) or as in (4.11.107) að +b

ξ 1 − ν f½ξ 2 − ða − bÞ2 ½ða + bÞ2 − ξ 2 gν − 1=2 Jν ðt ξÞ dξ

ja − b j

   pffiffiffi 1 ab ν Jν ðatÞJν ðb tÞ = 23 ν − 1 π Γ ν + 2 t

(4:11:109)

This integral can be reduced to more simpler forms ðβ α

ξ 1 − ν f½ξ 2 − α2 ½β2 − ξ 2 gν − 1=2 Jν ðt ξÞ dξ

!ν    2    pffiffiffi 1 β − α2 α+β β−α 3ν−1 Jλ π Γ ν+ =2 t Jλ t t 2 2 2 and

(4:11:110)

237

4.11 The Asymptotic Limit of the Bessel Function

ðα

ξ ν ðα2 − ξ 2 Þν − 1=2 Jν ðt ξÞ dξ

0 3ν−1

=2

(4:11:111)

   2 ν   2 pffiffiffi 1 α αt Jν π Γ ν+ t 2 2

The Struve transforms H(s,ν) are less numerous than the Hankel transforms G(s,ν). However, if used, they give interesting integrals of the Bessel functions of the second kind, as it is illustrated below. From the function transform pair gðtÞ = J2 ν + 1 ðαt1=2 Þ

; α>0

,

−

 2 α α Y ν+1 4s 2 s3=2  2 u α t3=2 αt H ;ν = − 3=2 Yν + 1 t 4u 2u

3 1 < Re ν < 2 4

Hðs;νÞ = −

(4:11:112)

we have ∞ ð

0

 2 1 αt 2 du = − 1=2 J2 ν + 1 ðα t1=2 Þ Yν ðuÞ Yν + 1 u 4u αt

(4:11:113)

For ν = −1/2, the above integral can be reduced to the trigonometric integral ∞ ð

  pffiffiffiffiffiffi 1 β π dx = J0 ð2 α βÞ sinðα xÞ cos x x 2

(4:11:114)

0

A very similar function transform pair gðtÞ =

1 J2 ν ðα t1=2 Þ t1=2

;

α>0 ,

− 1 < Re ν
−

3 2

1 2 e − α =2 s 2 s3=2

(4:11:118)

it follows from (4.11.93) that ∞ ð

2

e − α t=x Yν ðxÞ dx = 2Yν ðα t1=2 Þ Kν ðα t1=2 Þ x

(4:11:119)

0

In the final example, the Struve transform has the shifted Heaviside step function (      2 ) αt 2 αt 3 1=2 Jν=2 − Yν=2 gðtÞ = t ; − < Re ν < 1 2 2 2 (4:11:120) 4uðs − αÞ Hðs; λÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π sðs2 − α2 Þ and this leads to the integral with the limit of integration which is not at the origin (    ∞  2 ) ð Yν ðxÞ π t 2 t pffiffiffiffiffiffiffiffiffiffiffiffi dx = − Yν=2 (4:11:121) Jν=2 4 2 2 x2 − t 2 t

Finally, the Lamborn expression for the shifted Dirac delta function will be applied to the sequence of functions fn(t,λ). This sequence converges to the sum S(t,λ) Sðt, λÞ =

∞ X

fn ðt, λÞ

(4:11:122)

n

where λ denotes a parameter. In the same way as applying in (4.11.13), we have ∞ ð

Sðt, λÞ =

Sðtx, λÞ δðx − 1Þ dx 0

= lim

(

ν!∞

ν

"∞ ð ∞ X n

0

#) fn ðtx, λÞ Jν ðνxÞ dx

(4:11:123)

4.11 The Asymptotic Limit of the Bessel Function

239

It is assumed that the order of summation, integration and the limit of the series can be reversed and the integrals in (4.11.123) exist. If the sum in (4.11.123) is known then the expression (4.11.123) permits to determine the limit of sum of integrals. For example, from (4.11.34) ∞ ð

e− ξ

2 x2

Jν ðνxÞ dx =

pffiffiffi  2 ν π − ν2 =8 ξ 2 Iν=2 e 8ξ 2t

(4:11:124)

0

it follows that (  2 ) 2 2 ν e − ν =8 n t ν 2t 2 pffiffiffi lim Iν=2 = pffiffiffi e − n t 2 ν!∞ 8 nt π n pffiffiffi ξ =t n

(4:11:125)

but it is known that ∞ X

2

2

e−nt =

n =1

e− t 1 − e − t2

(4:11:126)

and therefore we have limit of the sum ! (  2 ) 2 2 2 ∞ X e − ν =8 n t ν 2t e− t pffiffiffi lim ν Iν=2 = pffiffiffi ν!∞ 8n t2 π 1 − e − t2 n n=1

(4:11:127)

Similarly from ∞ X

1 Jn ðλÞ Jn + 1 ðλÞ sin½ð2 n + 1Þ x = J1 ð2λ sin xÞ 2 n=0 ∞ X 1 ð − 1Þn Jn ðλÞ Jn + 1 ðλÞ cos½ð2 n + 1Þx = J1 ð2 λ cos xÞ 2 n=0

(4:11:128)

and considering that    sin λ sin − 1 ba pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðaxÞJλ ðb xÞ dx = ; b2 − a2 0 ∞    ð cos λ sin − 1 ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðaxÞJλ ðb xÞ dx = b 2 − a2 ∞ ð

0 −1 ∞ < X sin νsin ð2n + 1Þ ν = lim ν Jn ðλÞ Jn + 1 ðλÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 > ν!∞> ; : n=0 ν2 − ð2n + 1Þ νt

(4:11:130)

1 = J1 ð2λ sin tÞ 2 and 9 8    t > > −1 ∞ = < X cos νsin ð2 n + 1Þ ν lim ν ð − 1Þn Jn ðλÞJn + 1 ðλÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 > ν!∞> ; : n=0 ν2 − ð2n + 1Þ νt

(4:11:131)

1 = J1 ð2λ cos tÞ 2 As demonstrated in this section, mathematical operations performed with the order ν of the Bessel function Jν(νx) lead to many interesting results. Established asymptotic limits, representations of mathematical constants and functions, evaluated integrals and series undoubtedly prove that the Lamborn sequence for the shifted Dirac delta function δ(x – 1), is a very efficient tool in the theory of special functions.

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Appendix A The Bessel and Related Functions Integrals Miscellaneous integrals recently reported in the literature [158, 165–201], mainly from the period 1990–2017, are presented here and therefore they are not available in major tabulations of integrals of the Bessel and related functions.

A1 Integrals Containing One Bessel or Related Function, and Elementary and Special Functions ða 0 ∞ ð

Jν ðb xÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx = π xða − xÞ

a

ða

rffiffiffi     a ab ab Jν=2 + 1=4 Jν=2 − 1=4 2 2 2

    Jν ðb xÞ π ab ab pffiffiffiffiffiffiffiffiffiffiffiffiffi dx = − Jν=2 Yν=2 2 2 2 x2 − a2 xν ða2 − x2 Þ

ν − 1=2

;

;

a>0 ;

Re ν > −1

ν≥ −

1 2

    2 pffiffiffi 1 a2ν ab Jν ðb xÞ dx = 2 3ν − 1 π Γ ν + Jν 2 bν 2

0

(A:1:1)

(A:1:2)

(A:1:3)

1 Re ν > − 2 ðb

x1 − ν ½ðx2 − a2 Þðb2 − x2 Þ

ν −1=2

Jν ðc xÞ dx (A:1:4)

a

      ν pffiffiffi 1 ðb2 − a2 Þ ða + bÞc ðb − cÞ c 3ν − 1 Jν Jν =2 πΓ ν+ cν 2 2 2 ∞ ð

;

1 Re ν > − 2

pffiffiffiffiffiffiffiffiffi xα=2 −1 ð1 + xÞ1 − α − β Jν ðλ 1 + xÞ dx

0

!β − 1 !  Γ ν2 − β + 1 λ2 ν ν λ2  = 1 F2 1 − α; β − , β + ; − 4 4 2 2 Γ ν2 + β

(A:1:5)

    pffiffiffiffiffiffiffiffiffi λ λ Yν=2 Jν ðλ 1 + xÞ dx = π Jν=2 2 2

(A:1:6)

!  ΓðαÞ Γ β − ν2 − 1 ν ν λ2  1 F2 + − α − β + 2; − β + 2, 1 + ν; − 4 2 2 Γðν + 1Þ Γ α + β − ν2 − 1 ∞ ð

x

− 1=4

ð1 + xÞ

− 1=2

0

https://doi.org/10.1515/9783110681642-006

250

Appendix A The Bessel and Related Functions Integrals

∞ ð

         pffiffiffiffiffiffiffiffiffi π λ λ λ λ Yðν + 1Þ=2 + Jðν + 1Þ=2 Yðν − 1Þ=2 x − 3=4 Jν ðλ 1 + xÞ dx = − Jðν − 1Þ=2 2 2 2 2 2

0

(A:1:7) ∞ ð

0





pffiffiffiffiffiffiffiffiffi 1 ½x ð1 + xÞ − 1=2 J0 ðλ 1 + xÞ dx = 2 − J0 ðλÞ + π ½J0 ðλÞ H1 ðλÞ − J1 ðλÞ H0 ðλÞ λ (A:1:8)

∞ ð

 pffiffiffiffiffiffiffiffiffi J0 ðλÞ ð1 + xÞ − 1=2 J1 λ 1 + x dx = 2 λ

0 ∞ ð

0 ∞ ð

(A:1:9)

 pffiffiffiffiffiffiffiffiffi sin λ x − 3=4 ð1 + xÞ − 1=4 J1 λ 1 + x dx = 2 λ x

− 1=2

ð1 + xÞ

− 1=4

0

(A:1:10)

    pffiffiffiffiffiffiffiffiffi 2 1 Jν ðλ 1 + xÞ dx = pffiffiffi Jν − 1 ðλÞ Sν=2 ðλÞ + − ν Jν ðλÞ S − ν=2 ðλÞ 2 λ (A:1:11)

∞ ð

 pffiffiffiffiffiffiffiffiffi cos λ x − 3=4 J0 λ 1 + x dx = 2 λ

0

ða

2

x ð1 − 2x Þ e 2

0 ∞ ð

0

(A:1:12)

xλ ðx2 − a2 Þ2

−x2

ae− a ½ J1=2 ðaxÞ dx = − pffiffiffi π

Jν ðbxÞ dx =

2

(A:1:13)

  πaλ − 2 b πðλ + νÞ Jν + 1 ðabÞ tan 4 2   λ+ν−3   3 − λ  Γ b 5 − λ + ν 5 − λ − ν a2 b2 2  +  , ; 1 F2 2; 5−λ+ν 4 2 2 2 2Γ 2 − ðRe ν + 1Þ < Re λ
0

(A:1:16)

Appendix A The Bessel and Related Functions Integrals

∞ ð

lnð1 + x2 Þ J1 ða xÞ dx = 0 ∞ ð

0 ∞ ð

0

ða 0 ∞ ð

0 ∞ ð

0 ∞ ð

0 ∞ ð

0

2 K0 ðaÞ a

(A:1:18)

 ax  a pffiffiffiffiffiffi cosðbxÞ pffiffiffiffiffiffiffiffiffiffiffi cosh J dx = K ðbÞ berð2 abÞ 0 0 1 + x2 1 + x2 1 + x2

(A:1:19)

sin − 1

x a

J1 ðb xÞ dx =

 π h a i2 − J0 ða bÞ J0 b 2b 2

0 ∞ ð

0 ∞ ð

0

(A:1:20)

 2  2  h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i cos a4 ab e − x J0 ðbxÞ ber a ð1 + b2 Þx dx = pffiffiffiffiffiffiffiffiffiffiffi I0 4 1 + b2

(A:1:21)

 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii sin a4 ðÞ a2 b e − x J0 ðbxÞ bei a ð1 + b2 Þx dx = pffiffiffiffiffiffiffiffiffiffiffi I0 4 1 + b2

(A:1:22)

pffiffiffiffiffiffi 1 1 − Eα ðaα Þ pffiffiffi J1 ð2 a xÞEα ðxα Þ dx = pffiffiffi x a

(A:1:23)

  pffiffiffiffiffiffi 1 1 1 pffiffiffi J2 β −1 ð2 axÞEα, β ðxα Þ dx = pffiffiffi − Eα, β ðaα Þ x a ΓðβÞ

(A:1:24)

x J0 ðaxÞ Pn ð1 − 2x2 Þ dx =

0 ∞ ð

(A:1:17)

 ax  a pffiffiffiffiffiffi sinðbxÞ pffiffiffiffiffiffiffiffiffiffiffi sinh J0 dx = K0 ðbÞ beið2 abÞ 2 2 2 1+x 1+x 1+x

ð1 0 ∞ ð

251

xn e − x

2 =4

1 J2n + 1 ðaÞ a

(A:1:25)

2 =4

Jn ðaxÞ D2n − 1 ðxÞ dx = ð − 1Þn an − 1 e − a

D2n − 1 ðaÞ

2 =4b

x1 − ν Jν ða xÞ γðν, bx2 Þ dx = 21 − ν aν − 2 e − a

(A:1:27)

pffiffiffiffiffiffi x − ðα + 1Þ=2 Jα + 1 ð2 axÞ νðxÞ dx = − a − ðα + 1Þ=2 νða, αÞ ; 2 =x

e− a x

Yν ðxÞ dx = 2 Yν ðaÞKν ðaÞ

; Re ν > −

(A:1:26)

3 2

α > −1

(A:1:28)

(A:1:29)

252

Appendix A The Bessel and Related Functions Integrals

∞ ð

a

a i2  Yν ðxÞ π h a i2 h pffiffiffiffiffiffiffiffiffiffiffiffiffi dx = ; − Yν=2 Jν=2 4 2 2 x 2 − a2

ð1 x Y0 ðaxÞPn ð1 − 2x2 Þ dx = 0

ðα 0

−

3 < Re ν < 1 2

(A:1:30)

1 ½Sn + 1 ðaÞ + π Yn + 1 ðaÞ πa

(A:1:31)

  3 x2 xν + 2 Yν ðxÞ 2 F1 1, 2ν + ; ν + 2; 2 dx = α 2 (A:1:32)

 α α π3=2 Γ ν + 21 α2ν + 3 3  J Y ; − < Re ν < 0 ν=2 ν=2 3 ν + 1 2 2 4 2 Γ 2ν + 2  pffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffi ða x cosh b a2 − x2 sinh a 1 + b2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi I0 ðxÞdx = a2 − x 2 1 + b2 0 ∞ ð

K0 ðxÞ dx =

(A:1:33)

π 2

(A:1:34)

0 ∞ ð

x K0 ðxÞ dx = 1 0 ∞ ð

0 ∞ ð

0 ∞ ð

0 ∞ ð

0 ∞ ð

0 ∞ ð

0

   n+1 2 xn K0 ðxÞ dx = 2n − 1 Γ 2

(A:1:35)

;

n = 0, 1, 2, 3, ...

(A:1:36)

x 1 pffiffiffiffiffiffiffiffiffiffiffi K0 ðaxÞ dx = ½sin a CiðaÞ − cos a SiðaÞ 2 a 1+x

(A:1:37)

π ln x K0 ðxÞ dx = − ðln 2 + γÞ 2

(A:1:38)

e − x ln x K0 ðxÞ dx = −ðγ − ln 2Þ

(A:1:39)

pffiffiffi sinh x ln x K0 ðxÞ dx = 23=2 π ð4 − 5 ln 2 − γ − πÞ x3=2

(A:1:40)

pffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi − c b2 − a2 π e coshða xÞ K0 b c2 + x2 dx = pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 − a2

; b>a>0 ;

c>0

(A:1:41)

Appendix A The Bessel and Related Functions Integrals

∞ ð

0 ∞ ð

0 ∞ ð

 2   2   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π a b a J0 sinðb xÞ K0 ðxÞ bei a ð1 + b2 Þx dx = pffiffiffiffiffiffiffiffiffiffiffi sinh 4 4 2 1 + b2

(A:1:42)

 2   2   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π a b a J0 cosðbxÞ K0 ðxÞ ber a ð1 + b2 Þ x dx = pffiffiffiffiffiffiffiffiffiffiffi cosh 2 4 4 2 1+b

(A:1:43)

e 0 ∞ ð

0 ∞ ð

0 ∞ ð

253

−x

 2  2    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a a b 2 I0 J0 ðb xÞ ber a ð1 + b Þx dx = pffiffiffiffiffiffiffiffiffiffiffi cos 2 4 4 1+b

(A:1:44)

 2  2    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a a b I0 e −x J0 ðb xÞ bei a ð1 + b2 Þx dx = pffiffiffiffiffiffiffiffiffiffiffi sin 4 4 1 + b2

(A:1:45)

 pffiffiffiffiffiffi e − x I2 n ðx sin θÞ bei4 n 2 cos θ αx dx = ð − 1Þn sec θ sin α I2 n ðα sin θÞ

(A:1:46)

pffiffiffiffiffiffi e −x I2 n + 1 ðx sin θÞ ber4 n + 2 ð2 cos θ α xÞ dx

(A:1:47)

0

= ð−1Þn + 1 sec θ sin α I2 n + 1 ðα sin θÞ ∞ ð

 2   pffiffiffi a 2α + 1 pffiffiffi −x2 =a x μðx, β, αÞ dx = 2β π a μ , β. K1=4 xe a 8 4

0

(A:1:48)

α, β > −1 ∞ ð

 3=2    pffiffiffi pffiffiffi 2x α β+1 p ffiffiffiffiffiffiffi ffi μðx, β, αÞ dx = 3 π a μ a, β. x K1=3 3 27a

(A:1:49)

 3=2    2x α−2 x3=2 K1=3 pffiffiffiffiffiffiffiffi μðx, β, αÞ dx = 3 β + 2 π a 3=2 μ a, β. 3 27a

(A:1:50)

0 ∞ ð

0 ∞ ð

 x K2=3

0 ∞ ð

0

∞ ð

a

   2 x3=2 α−1 β + 3=2 pffiffiffiffiffiffiffiffi μðx, β, αÞ dx = 3 π a μ a, β. 3 27a

 2 x μðx, β, αÞ dx = a      pffiffiffi a 2α + 3 a 2α − 1 2β − 1 π a μ , β. + 4μ , β. 8 4 8 4

x3=2 e −x

2 =a

pffiffiffi Kν ð xÞ ða − xÞ

μ+1

(A:1:51)

K3=4

dx =

pffiffiffi Γð − aÞ Kμ + ν a 2 μ aðμ + νÞ=2

(A:1:52) ;

α, β > −1

(A:1:53)

254

Appendix A The Bessel and Related Functions Integrals

∞ ð

x n − 1=2 e −x

0 ∞ ð

x

2=4

Kn + 1=2 ðaxÞ D2n ðxÞ dx = ð−1Þn

n + 3=2 −x2=4

e

rffiffiffi π 2=4 Γð2nÞ an − 1=2 e a D−2n ðaÞ 2

(A:1:54)

rffiffiffi π 2=4 Kn + 1=2 ðaxÞ D2n + 1 ðxÞ dx = ð−1Þ Γð2n + 3Þ an − 1=2 e a D−ð2n + 3Þ ðaÞ 2 n

0

(A:1:55) ∞ ð



0 ∞ ð

0 ∞ ð

0

x ν + 1=2 π bν − 1=2 Hν − 1=2 ðbxÞ dx = ν + 1=2 ½I0 ðabÞ − L0 ðabÞ 2 2 a +x 2 Γðν + 1Þ

xν πaν − 1 H ðbxÞdx = ½ L − ν ðabÞ − Lν ðabÞ  ν x 2 + a2 2 sinðπ νÞ

;

(A:1:56)

j Re ν j < 1

xν + 2k + 1 π aν + 2k secðπ νÞ Hν ðbxÞdx = ð−1Þk ½ I − ν ðabÞ − Lν ðabÞ  2 2 x +a 2     3 3 1 < Re ν < min − k+ − 2k, − k ; 2 2 2

(A:1:57)

k = 0, 1, 2 (A:1:58)

∞ ð

0

xν πaν − 1 H ðbxÞdx = ½ cscðπ νÞ H −ν ðabÞ − cotðπ νÞ Hν ðabÞ  ν x 2 − a2 2

(A:1:59)

j Re ν j < 1 ∞ ð

0

xν + 2k + 1 π aν +2k Hν ðbxÞdx = ½ secðπ νÞ J− ν ðabÞ + tanðπ νÞ Hν ðabÞ  2 2 x −a 2     3 3 1 − k+ < Re ν < min − 2k, − k ; k = 0, 1, 2 2 2 2 (A:1:60)

∞ ð

0

xν + 1 ðx2 + a2 Þλ + 1

Hν ðbxÞdx =

ð1 x E0 ðaxÞPn ð1 − 2x2 Þ dx = 0 ∞ ð

0

πaν − λ bλ sec½π ðλ − νÞ ½ I λ − ν ðabÞ − Lν − λ ðabÞ  2λ + 1 Γðλ + 1Þ   3 1 3 − < Re ν < min Re λ + , 2 Re λ + 2 2 2

(A:1:61)

1 2 E2n + 1 ðaÞ − a ð2n + 1Þ π a

(A:1:62)

 pffiffiffiffiffiffi ber 2 ax π  pffiffiffiffiffiffi dx = J0 2 ab 2 2 b +x 2b

(A:1:63)

255

Appendix A The Bessel and Related Functions Integrals

∞ ð

e 0 ∞ ð

0 ∞ ð

"pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # a2 + b2 − a 1 Ji0 ðbxÞdx = ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a a2 + b2 + a

(A:1:64)

 pffiffiffiffiffiffi sin a ciðaÞ siðxÞ Ji0 2 ax dx = − 2a 2

(A:1:65)

 pffiffiffiffiffiffi 1 − cos a siðaÞ ciðxÞ Ji0 2 a x dx = − 2a 2

(A:1:66)

pffiffiffi  pffiffiffiffiffiffi 1 π 2 sin a + siðaÞ pffiffiffi siðxÞ Ji1 2 a x dx = pffiffiffi + pffiffiffi + 4C a x 2 a a

(A:1:67)

 pffiffiffiffiffiffi pffiffiffi 1 2 + γ + ln a + 2 cos a − ciðaÞ pffiffiffi ciðxÞ Ji1 2 ax dx = pffiffiffi + 4 Sð aÞ x a

(A:1:68)

0 ∞ ð

0 ∞ ð

− ax

0 ∞ ð

e − a x Ji0 ðxÞ dx = −

0 ∞ ð

e

−ax

0 ∞ ð

e 0

" # 1 1 Jiν ðxÞ dx = pffiffiffiffiffiffiffiffiffiffiffi − 1 aν ða + a2 + 1Þν

e − a x Yi0 ðxÞ dx =

0 ∞ ð

1  pffiffiffiffiffiffiffiffiffiffiffi ln a + a2 + 1 a

−ax

(A:1:69)

;

Re ν > 0

h  pffiffiffiffiffiffiffiffiffiffiffi i2 ln a + a2 + 1 πa

(A:1:70)

(A:1:71)

" # ν i cotðπ νÞ cscðπ νÞ h pffiffiffiffiffiffiffiffiffiffiffi 1 Yiν ðxÞ dx = a + a2 + 1 − 1 − 1− pffiffiffiffiffiffiffiffiffiffiffi ν aν aν ða + a2 + 1Þ

jRe νj < 1 (A:1:72) ∞ ð

0 ∞ ð

0

e − a x Ki0 ðxÞ dx = −

i2 π2 1 h  pffiffiffiffiffiffiffiffiffiffiffi − ln a + a2 + 1 8a 2a

(A:1:73)

2

3 π ν  cos p ffiffiffiffiffiffiffiffiffiffiffi ν π cscðπ νÞ 6 1 7 2 e − a x Kiν ðxÞ dx = − 4 a + a2 + 1 +  pffiffiffiffiffiffiffiffiffiffiffi ν 5 − aν 2a ν 2 a+ a +1

jRe ν < 1j (A:1:74)

256

Appendix A The Bessel and Related Functions Integrals

A2 Integrals Containing Product of Two Bessel or Related Functions, and Elementary and Special Functions ðτ

rffiffiffiffiffiffiffiffiffiffi h h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h pffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ab τ J1 2 ða + bÞ τ J0 2 a ðτ − xÞ J0 2 b x dx = a+b

0 ∞ ð

e − cx J0 ðaxÞ J0 ðbxÞ dx =

0

(A:2:1)

2 FðkÞ πβ

ð1

dx α pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k = ; 2 2 2 β ð1 − x Þð1 − k x Þ 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α= ða + bÞ2 + c2 − ða − bÞ2 + c2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 β= ða + bÞ2 + c2 + ða − bÞ2 + c2 2

0 < k2 < γ < 1

FðkÞ =

∞ ð

xe

− cx

2c

J0 ðaxÞ J0 ðbxÞ dx =

2 EðkÞ − FðkÞ 1 − k2

π a β3 ð1 − k2 Þ

0

;

γ=

α2 b2

(A:2:2)

 (A:2:3)

ð1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − k 2 x2 dx EðkÞ = 1 − x2 0

π=2 ð

J0 ða cos θÞ J1 ða sin θÞ cos θ d θ = 0 ∞ ð

0

Πðγ, kÞ = 0 ∞ ð

0

(A:2:4)

2c fFðkÞ − Πðγ, kÞg πaβ

e − cx J1 ðaxÞ J0 ðbxÞ dx = − ð1

pffiffiffi i 1h J0 ðaÞ − J0 ð 2 aÞ a

(A:2:5) dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 − γx Þ ð1 − x2 Þð1 − k2 x2 Þ

x e − cx J0 ðaxÞ J1 ðbxÞ dx =

2 π b β3 ð1 − k2 Þ

2 2 2  2  β − b2 FðkÞ − a + c − b EðkÞ 2 1−k



(A:2:6) ∞ ð

x2 e − cx J0 ðaxÞ J1 ðbxÞ dx =

2 cð7α2 + β2 − 5 b2 − 5b2 k2 Þ π b β5 ð1 − k2 Þ3

0

+

FðkÞ

2 c ½8b2 ð1 + k2 Þ − α2 k2 − 14α2 − β2  π b β5 ð1 − k2 Þ4

(A:2:7) EðkÞ

257

Appendix A The Bessel and Related Functions Integrals

∞ ð

0 ∞ ð

e − cx 2 J0 ðaxÞ J1 ðbxÞ dx = x π bβ

  πcβ + β2 EðkÞ − β2 − b2 FðkÞ + c2 Πðγ, kÞ 2

x e − cx J0 ðaxÞ J2 ðbxÞ dx =

2

2 c 2 ðb − 2β Þ 4β 4c + EðkÞ + FðkÞ + Πðγ, kÞ π b2 β b2 π b2 β π b2

e − cx J0 ðaxÞ J2 ðbxÞ dx = −

2 2c + b2 π β3 ð1 − k2 Þ h

x2 e − cx J0 ðaxÞ J0 ðbxÞ dx =

2

(A:2:9)  2 4c FðkÞ − EðkÞ − Πðγ, kÞ 1 − k2 π b2 β

2 2

2 β2 ð1 − k Þ − 5 c2 − 3c2 k2

2 2

2

−

(A:2:10)

i FðkÞ

π β5 ð1 − k2 Þ3

0

∞ ð

(A:2:11)

4 ½β ð1 − k Þ − 4c ð1 + k Þ 2

2

π β5 ð1 − k2 Þ4

EðkÞ

e − cx J2 ðaxÞ J0 ðbxÞ dx x

0

 c  2 = ð3β + b2 − 2a2 − 2 c2 ÞFðkÞ − 3β2 EðkÞ + ða2 − 2b2 + 2c2 ÞΠðγ, kÞ 2 πa β ∞ ð

0



(A:2:8)

0

∞ ð

−

2

0 ∞ ð

(A:2:12)

e − cx 1 J2 ðaxÞ J0 ðbxÞ dx = fβ2 ð8a2 + 11 c2 − 6 b2 ÞEðkÞ x2 9π a2 β − ½ β2 ð2β2 + 3c2 Þ − 6α2 ða2 + c2 Þ

(A:2:13)

+ b2 ð6 α2 + 4a2 + 9 c2 − 6 b2 ÞFðkÞ + c2 ð9 b2 − 9 a2 − c b2 ÞΠðγ, kÞg ∞ ð

0 ∞ ð

0

2

x e − a x J0 ðxÞY0 ðxÞ dx =

  e − 1=2a 1 K0 2πa 2a

(A:2:14)

½ J0 ðaxÞ J0 ðbxÞ − Y0 ðaxÞ Y0 ðbxÞ 2 dx = − K0 ðacÞK0 ðbcÞ c 2 + x2 πc

a, b, c > 0 ∞ ð ½ J0 ðaxÞ J0 ðbxÞ − Y0 ðaxÞ Y0 ðbxÞ dx x2 − c2

(A:2:16)

0

π =− ½J0 ðacÞ Y0 ðbcÞ + J0 ðbcÞ Y0 ðacÞ 2c

(A:2:15)

;

a, b, c > 0

258

Appendix A The Bessel and Related Functions Integrals

π=2 ð

J0 ða cos θÞ I1 ða sin θÞ cos θ dθ = 0 ∞ ð

0 ∞ ð

0 ∞ ð

pffiffiffi i 1h 1 − J0 ð 2 aÞ a

(A:2:17)

 a  ax  pffiffiffiffiffiffi x e−b pffiffiffiffiffiffiffiffiffiffiffi sin ðbxÞI J dx = bei 2 ab 0 0 b 1 + x2 1 + x2 1 + x2

(A:2:18)

 a  ax  pffiffiffiffiffiffi x e−b pffiffiffiffiffiffiffiffiffiffiffi cos J0 ðbxÞI0 dx = ber 2 ab 2 2 b 1+x 1+x 1 + x2

(A:2:19)

x e − 2x J0 ðaxÞ K0 ðaxÞ dx =

  2  2  π a a − Y0 H0 4 4 16

(A:2:20)

x ½J0 ðaxÞ2 Pn ð1 − 2x2 Þ dx =

n o 1 ½Jn ðaÞ2 + ½Jn + 1 ðaÞ2 2 ð2n + 1Þ

(A:2:21)

2

0

ð1 0

ð1

 pffiffiffiffiffi  pffiffiffiffiffi x J0 2 ax K0 2 ax Pn ð1 − 2x2 Þ dx (A:2:22)

0

  pffiffiffi  pffiffiffi  pffiffiffi  pffiffiffi  1 = J2n 2 a K2n 2 a + J2n + 2 2 a K2n + 2 2 a 2 ð2n + 1Þ ð1 x ½J0 ðaxÞ2 Pn ð1 − 2x2 Þ dx = 0 ∞ ð

0

n o 1 ½Jn ðaÞ2 + ½Jn + 1 ðaÞ2 2 ð2n + 1Þ

(A:2:23)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i α ffi e − cx 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − α2 + a2 sin − 1 p ffiffiffiffiffiffi J ðaxÞ J ðbxÞ dx = α a + 2 c b2 − α2 − b 1 1=2 a x3=2 a πb

Re c > jImða ± bÞj (A:2:24) ∞ ð

0 ∞ ð

0 ∞ ð

0 ∞ ð

0

rffiffiffiffiffiffiffiffiffiffiffih pffiffiffiffiffiffiffiffiffiffiffiffiffiffii e − cx 2 pffiffiffi J1 ðaxÞ J1=2 ðbxÞ dx = b − b2 − α 2 π a2 b x

(A:2:25)

pffiffiffi − cx J1 ðaxÞ J1=2 ðbxÞ dx = xe

rffiffiffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2 α a2 − α2 2 πa b β2 − α2

(A:2:26)

pffiffiffi − cx J1 ðaxÞ J3=2 ðbxÞ dx = xe

rffiffiffiffiffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 2 α2 b2 − α2 2 3 πa b β2 − α2

(A:2:27)

e − cx pffiffiffi J1 ðaxÞ J3=2 ðbxÞ dx = x

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 −1 ε 2 a sin − α a2 − α2 2 π a2 b3 a

(A:2:28)

259

Appendix A The Bessel and Related Functions Integrals

∞ ð

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   α  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 2a2 α 2 − α2 + pffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 3sin − 1 α a 2 π a2 b5 a a2 − α2

e − cx pffiffiffi J1 ðaxÞ J5=2 ðbxÞ dx = x

0

(A:2:29) π=2 ð

0 ∞ ð

0 ∞ ð

pffiffiffi h pffiffiffi i 2 pffiffiffi J1 ða sin θÞ J1 ða cos θÞ dθ = 2 J1 ðaÞ − J1 ð 2 aÞ a

e − cx J1 ðaxÞ J1 ðbxÞ dx = −

2β fFðkÞ − EðkÞg π ab

x e − cx J1 ðaxÞ J1 ðbxÞ dx =

2c π a b βð1 − k2 Þ

0 ∞ ð

x2 e − cx J1 ðaxÞ J1 ðbxÞ dx =

(A:2:31)

1 + k2 EðkÞ − FðkÞ 1 − k2

h i 2 2 c2 + 7c2 k2 − β2 ð1 − k2 Þ π a b β3 ð1 − k2 Þ3

0

−

(A:2:30)



FðkÞ

h i 2 2 β2 ð1 + k2 Þð1 − k2 Þ − c2 ð1 + 14 k2 + k4 Þ π a b β3 ð1 − k2 Þ4

EðkÞ (A:2:32)

∞ ð

0

 e − cx c  2 J1 ðaxÞ J1 ðbxÞ dx = β EðkÞ − ðβ2 + b2 ÞFðkÞ − ða2 − b2 Þ Πðγ, kÞ x π ab β (A:2:33)

∞ ð

0

e − cx 1 J1 ðaxÞ J1 ðbxÞ dx = f½β2 ð4α2 + c2 − 2 a2 − 2b2 Þ + 3b2 c2 FðkÞ x2 3 π abβ (A:2:34))

+ ½β2 ð2 a2 + 2 b2 − c2 Þ  EðkÞ + 3 c2 ða2 − b2 ÞΠðγ, kÞg − ∞ ð

xe 0 ∞ ð

0 ∞ ð

− a x2

(

xe 0

  1 e − 1=2a 1 J1 ðxÞY1 ðxÞ dx = − + K1 2πa π 2a

a 2cβ e − cx J1 ðaxÞ J2 ðbxÞ dx = 2 + b π ab2 − cx

ac 2b

( EðkÞ − FðkÞ −

a2 β2

) Πðγ, kÞ

(A:2:35)

ð2 β − α2 − b2 Þ ½b2 ðb2 + c2 − a2 Þ − 2ðβ2 − α2 Þ EðkÞ J2 ðaxÞ J1 ðbxÞ dx = FðkÞ − 2 2 π ab βð1 − k Þ π ab2 β3 ð1 − k2 Þ

)

(A:2:36)

260

Appendix A The Bessel and Related Functions Integrals

∞ ð

2c ½2β2 ð1 − k2 Þ + 7 b2 k 2 + b2 − 5α2 − 3α2 k 2 

x2 e − cx J1 ðaxÞ J2 ðbxÞ dx =

FðkÞ

π ab 2 β3 ð1 − k2 Þ3

0

+

2 c ½2ð1 + k2 Þðβ2 − 6 α2 + α2 k 2 Þ + b 2 ð1 + 14k2 + k4 Þ π ab 2 β3 ð1 − k2 Þ4

EðkÞ

(A:2:37) ∞ ð

0

e − cx ac 2a c2 J1 ðaxÞ J2 ðbxÞ dx = − 2 + Πðγ, kÞ x π b2 β b

 2β  2 ð2 a − b2 − c2 ÞEðkÞ + ðα2 − 2a2 + b2 + c2 ÞFðkÞ 2 3 π ab (A:2:38) qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 3=2 ∞ ð β2 − b2 pffiffiffi − cx 2a b (A:2:39) J2 ðaxÞ J3=2 ðbxÞ dx = xe π β4 ðβ2 − α2 Þ +

0

∞ ð

0 ∞ ð

0

∞ ð

0 ∞ ð

e − cx pffiffiffi J2 ðaxÞ J3=2 ðbxÞ dx = x

# rffiffiffiffiffiffiffiffi" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 b2 − α2 ðb2 − α2 Þ 2b3 2 + − π a4 3 3b3 b

(A:2:40)

e − cx a ð2 b2 + 4c2 − a2 Þ cð5b2 + 2c2 − 13 a2 Þ J1 ðaxÞ J2 ðbxÞ dx = + EðkÞ 2 x 8b2 12 π bα

e − cx J2 ðaxÞ J2 ðbxÞ dx =

−

c ½α2 ð13a2 − 5α2 − 5b2 − 2c2 Þ − 3b4  FðkÞ 4π ab2 β

+

c ½a2 ð2 b2 + 4c2 − a2 Þ − b4  Πðγ, kÞ 4π a2 β

 2β  2 ð2β − α2 ÞFðkÞ − 2 ða2 + b2 + c2 ÞEðkÞ 2 2 3π a b

x e − cx J2 ðaxÞ J2 ðbxÞ dx =

0

(A:2:41)

(A:2:42)

 2cβ 2 ð1 − k2 + k4 Þ 2 Þ FðkÞ EðkÞ − ð2 − k π a2 b2 ð1 − k2 Þ ð1 − k2 Þ (A:2:43)

∞ ð

h

2 − cx

xe

J2 ðaxÞ J2 ðbxÞ dx =

0

+

2

2 ð1 − k2 Þ ð2c2 + 2β2 − α2 Þ − c2 k2 ð5 + 3k2 Þ π a2 b 2 βð1 − k2 Þ3

i FðkÞ

h i 2 4 4c2 k2 ð1 + k2 Þ − ð1 − k2 Þ ðβ2 − α2 + α2 k 2 + c2 + c2 k 2 Þ π a2 b 2 βð1 − k2 Þ4

EðkÞ

(A:2:44)

261

Appendix A The Bessel and Related Functions Integrals

∞ ð

0

e − cx a2 c + J2 ðaxÞ J2 ðbxÞ dx = fβ2 ð5a2 + 5b2 + 2 c2 ÞEðkÞ x 4b2 6 π a2 b2 β

(A:2:45)

− ½β2 ð4a2 + 4b2 + c2 + β2 Þ + 3b4 FðkÞ + 3 ðb4 − a4 ÞΠðγ, kÞg ∞ ð

e − cx J2 ðaxÞ J2 ðbxÞ dx x2

0

 2 2 2 a2 c 1 + c β ð c + β2 Þ + 3b4 ð c2 + 4α2 − 4b2 Þ 4b2 30π a2 b2 β  + 4β2 ðα2 a2 + 2 a2 c2 − 2 b2 β2 − 2 a4 − b4 Þ FðkÞ + 15c2 ða4 − b4 ÞΠðγ, kÞ    + β2 ðα2 + β2 Þð8a2 + 8b2 − 2c2 Þ − 3βð8 a2 b2 + 5b2 c2 + 5 a2 c2 Þ EðkÞ

=−

∞ ð

0 ∞ ð

  2 e − 1=2a 1 K2 ð1 − 2aÞ − 2πa π 2a

2

x e − a x J2 ðxÞY2 ðxÞ dx = −

x3 e −x

2 =a

J2 ðxÞ Y2 ðxÞ dx = −

0

4 a2 ða + 2Þ e − a=2  a K0 + 4π π 2

að8 + 4a + a2 Þ e − a=2  a K1 + 4π 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ∞  ð − cx e 2 pffiffiffi J3 ðaxÞ J1=2 ðbxÞ dx = 3a2 b − 4 b3 + 12 b c2 9 π a6 b x 0

∞ ð

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  − b2 − α2 12β2 − 16b2 + 4α2 − 3α2 pffiffiffi cx x e J3 ðaxÞ J3=2 ðbxÞdx ¼

0

∞ ð

(A:2:46)

rffiffiffiffiffiffiffi " 2b3 4 π a3

2 − 3

(A:2:47)

(A:2:48)

(A:2:49)

! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 b2 − α2 ðb2 − α2 Þ þ 3b3 b

qffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a β2 − a2 5 − 3 2 β ðβ − α2 Þ e − cx J3 ðaxÞ J3=2 ðbxÞ dx = x3=2

0

(A:2:50)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 rffiffiffiffiffiffiffiffiffiffiffi 8 2 4 2 2 4 4 2 2 b3 < β − b ½4b ð2 b − α Þ − α  − 8b c= ; b4 9πa6 : (A:2:51)

∞ ð

0

2

x e − a x J3 ðxÞY3 ðxÞ dx = −

    3 16a 32a2 e − 1=2a 1 + K3 1− + 3 2πa π 3 2a

(A:2:52)

262

Appendix A The Bessel and Related Functions Integrals

∞ ð

  pffiffiffi 1 1 Jν ðaxÞ J2ν ð2 xÞ dx = Jν a a

; ν>0

(A:2:53)

0

  μ+ν−λ+1 ν b Γ 1 2   Jμ ðaxÞ Jν ðb xÞ dx = I μ − ν + λ + 1 ν−λ + 1 xλ λ a 2 Γðν + 1ÞΓ 0 2   ν−μ−λ+1 ν+μ−λ+1 b2 I = 2 F1 , ;ν + 1; 2 a 2 2 ∞ ð

Reðν + μ − λÞ > − 1 ∞ ð

; 0c

− ðk + 1Þ < Re λ < Re ν − 2k + 1

;

; k = 0, 1, 2, . . .

b=c (A:2:62)

Appendix A The Bessel and Related Functions Integrals

∞ ð

263

x − μ Jλ ðaxÞ Jμ ðaxÞ dx =

0

   aμ − ν − 1 bν Γ λ + ν −2 μ + 1 ν−λ−μ+1 λ+ν−μ+1 b2  2 F1 , ; ν + 1; 2 a 2 2 2μ Γðν + 1ÞΓ λ − ν +2 μ + 1

Reðλ + ν − μÞ > − 1

(A:2:63)

; 0 − 4

x μ Jν ðaxÞ Kν − μ + 1 ðb xÞ dx =

0

Re ν > −1 ∞ ð

0 ∞ ð

ν≥−

1 2

;

;

−ax

(A:2:68)

(A:2:69)

Re a > 0

2μ − 1 ΓðμÞ aν b μ − ν − 1 ða2 + b2 Þμ

(A:2:70)

1 Reð2μ − νÞ > 2

pffiffiffiffiffiffiffiffiffiffiffiffi ν     a4 + 1 − 1 x x pffiffiffiffiffiffiffiffiffiffiffiffi x Jν pffiffiffi Kν pffiffiffi J2ν ðaxÞ dx = 2 2 a2ν a4 + 1 e

0

;

  Γ λ + ν + 21 Γ ν − λ + 21 aλ − ν − 1  pffiffiffi Jν + λ ð2axÞ H2λ ð2 axÞ dx = 4 π Γðν − λ + 1Þ Γ 2 λ + 21

Reðν ± λÞ > −1Þ ∞ ð

1 2a

  2  2  pffiffiffi pffiffiffi 1 b b − νS − 1, ν Jν ðb xÞ Kν ðb xÞ dx = S0, ν 2a 2a 2a

(A:2:71)

(A:2:72)

264

Appendix A The Bessel and Related Functions Integrals

ða x 3 F2 0

  3 λ λ+1 νaλ − 2 e − a , , ; 1 − ν, 1 + ν; − 4x2 ½Jν ðaxÞ2 dx = − 2 ΓðλÞ 2 2 2

1 ν≥ ; λ>1 2   ða 3 1 a2ν − 1 e − a x 2 F1 , ν + ; 1 − ν; − 4x2 ½Jν ðaxÞ2 dx = − 4Γð2νÞ 2 2

(A:2:73)

(A:2:74)

0

ða 0

  3 3 ν a2ν e − a 2 x 2 F1 , ν + ; 1 − ν; − 4x ½Jν ðaxÞ2 dx = − 2 Γð2ν + 2Þ 2 2 

ða x 2 F2 0

 2 3 λ λ+1 νaλ − 2 e − a   , ; ; 1 − ν, 1 + ν; − x2 ½Jν ðaxÞ2 dx = − λ 2 2 2 Γ 2

1 ν≥ ; λ>1 2   ða 2 3 a2ν e − a x 1 F1 ; 1 − ν; − x2 ½Jν ðaxÞ2 dx = − ΓðνÞ 2

(A:2:75)

(A:2:76)

(A:2:77)

0

ða 0

ða

  2 1 a2ν e − a 2 x e −x 1 F1 − − ν; 1 − ν; − x2 ½Jν ðaxÞ2 dx = − ΓðνÞ 2

(A:2:78)

  2 3 1 e−a 2 x 2 F1 , 1; ; − 4x ½J1=2 ðaxÞ2 dx = − 4 2 2

(A:2:79)

0 ∞ ð

0

xν − k Jν + k ð2 bxÞ s2k, 2λ ð2 axÞ dx =

  22k − 3 Γ λ + ν + 21 Γ ν − λ + 21 a2λ I Γðν − k + 1Þ b 2λ + ν − k + 1

  1 1 a2 I = 2 F1 λ + ν + , λ − k + ; ν − k + 1; 1 − 2 b 2 2   1 ; Reðν + kÞ > −1 ; Re b > 0 Re ν ± λ; k ± λÞ > − 2 ∞   ð 22λ − 3 Γ λ + ν + 21 Γ ν − λ + 21 ν−k x Jν + k ð2 axÞ s2k, 2λ ð2axÞ dx = Γðν − k + 1Þ aν − k + 1 0   1 ; Reðν + kÞ > − 1 ; Re a > 0 Re ν ± λ; k ± λÞ > − 2

(A:2:80)

(A:2:81)

Appendix A The Bessel and Related Functions Integrals

∞ ð

0

  Γ λ + ν + 21 Γ ν − λ + 21 a2λ  xν − λ Jν + λ ð2bxÞ H2λ ð2axÞ dx = pffiffiffi 4 π Γðν − λ + 1Þ Γ 2λ + 21 b λ + ν + 1  2 F1

Reðν ± λÞ > − 1 ∞ ð

Yν ðaxÞ Yν 0 ∞ ð

0

265

;

Reλ > −

1 4

1 1 a2 λ + ν + , ; ν − λ + 1; 1 − 2 b 2 2

; Re b > 0

;



(A:2:82)

arg a < π

  b 1  pffiffiffiffiffiffi 5 dx = − J2ν 2 ab − 1 < Re ν < x a 4

;

a, b > 0

   pffiffiffiffiffiffi 3 1 a b 1 1 dx = − pffiffiffiffiffiffi J2ν + 1 2 ab − < Re ν < Yν Yν + 1 x x x 2 4 ab

(A:2:83)

;

a, b > 0 (A:2:84)

rffiffiffiffiffiffiffiffiffiffi h ðτ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h pffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ab τ I1 2 ða + bÞτ I0 2 aðτ − xÞ I0 2 b x dx = a+b

(A:2:85)

0

π=2 ð

0 ∞ ð

0 ∞ ð

0 ∞ ð

pffiffiffi h i pffiffiffi pffiffiffi 2 I1 ða sin θÞ I1 ða cos θÞ dθ = I1 ð 2 aÞ − 2 I1 ðaÞ a

ðln xÞ2 π2 1 I1 ðxÞK0 ðxÞ dx = 6 + + ðγ + 3 ln 2Þ2 + ðγ + 3 ln 2Þ x 6 14

(A:2:87)

ðln xÞ2 3 π2 1 1 I2 ðxÞK0 ðxÞ dx = + + ðγ − ln 2Þ2 − ðγ − ln 2Þ x 8 24 4 2

(A:2:88)

ðln xÞ3 3 π2 1 1 I2 ðxÞK0 ðxÞ dx = + + ζ ð3Þ − ðγ − ln 2Þ3 x 4 8 4 4

0

∞ ð

 2  3 π 9 ðγ − ln 2Þ + ðγ − ln 2Þ2 − + 8 8 4 2=a

x5 e − x

I3 ðxÞ K3 ðxÞ dx = −32 +

0

4 ea=2 a2 ð32 − 16a + 5 a2 − a3 Þ a + K0 π 8 2

ea=2 a ð128 − 64a + 25a2 − 6 a3 + a4 Þ a K1 + 8 2 ∞ ð

0

(A:2:86)

x e −x

2 =a

I3 ðxÞK3 ðxÞ dx = −

ð32 + 16a + 3a2 Þ ae a=2 a + K3 4 2 a2 2

(A:2:89)

(A:2:90)

(A:2:91)

266

Appendix A The Bessel and Related Functions Integrals

∞ ð

0 ∞ ð

0

ðτ 0

ln x 5 1 I4 ðxÞK0 ðxÞ dx = − ðγ − ln 2Þ x 32 16

(A:2:92)

ln x 194 1 I5 ðxÞK2 ðxÞ dx = − ðγ + 3 ln 2Þ x 735 21

(A:2:93)

  Γ α + 21 Γ α + 21 τ α + β + 1=2 pffiffiffiffiffiffiffiffiffi I α + β + 1=2ðτaÞ x ðτ − xÞ Iα ða xÞIβ ½aðτ − xÞ dx = 2 π a Γðα + β + 1Þ β

α

Re α, β > −

1 2

;

τ>0 (A:2:94)

ðτ 0



 Γ α + 21 Γ α + 32 τ α + β + 3=2 β+1 α pffiffiffiffiffiffiffiffiffi Iα + β + 1=2 ðτaÞ x ðτ − xÞ Iα ðaxÞIβ ½aðτ − xÞ dx = 2 π a Γðα + β + 2Þ

Re α > −

1 2

;

Re β > −

3 2

; τ>0 (A:2:95)

ðτ

h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h pffiffiffiffiffiffii  τ ðμ + νÞ=2 ðτ − xÞμ=2 xν=2 Iμ 2 a ðτ − xÞ Iν 2 b x dx = aμ=2 bν=2 a+b 0 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Iμ + ν + 1 2 ða + bÞτ

∞ ð

½K0 ðxÞ2 dx =

3 ζ ð2Þ π2 = 4 2

0 ∞ ð

K0 ðjxjÞ K0 ðja − xjÞ dx = −∞ ∞ ð

K0 ðxÞ K0

(A:2:96)

(A:2:97) π2 − a ; a>0 e 2

(A:2:98)

pffiffiffiffiffiffiffiffiffiffiffiffiffi π2 a2 + x2 dx = ½1 − a K0 ðaÞL − 1 ðaÞ + K1 ðaÞL0 ðaÞ 4

0

(A:2:99)

a>0 ∞ ð

−∞ ∞ ð

pffiffiffiffiffiffiffiffiffiffiffiffiffi K1 ðb b2 + x2 Þ π pffiffiffiffiffiffiffiffiffiffiffiffiffiffi K0 ðjx − ajÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dx = K0 b2 + a2 b b2 + x2

½Kν ðaxÞ2 dx =

π2 4a cosðπνÞ

x ½K0 ðxÞ2 dx =

1 2

0 ∞ ð

0

;

jRe νj
0

;

b, a real

(A:2:100)

(A:2:101)

(A:2:102)

267

Appendix A The Bessel and Related Functions Integrals

∞ ð

0 ∞ ð

0 ∞ ð

xn ½K0 ðxÞ2 dx = 2n − 1

pffiffiffi  n + 1 3 π Γ 2  4Γ n2 + 1

;

n = 0, 1, 2, 3, ...

(A:2:103)

 pffiffiffi ½Γ 43 4 2 x ln x ½K0 ðxÞ dx = − pffiffiffiffiffi ð2 + 3 ln 2 + γ + πÞ 2π

(A:2:104)

x ln x ½K0 ðxÞ2 dx = − ð1 − ln 2 + γÞ

(A:2:105)

1 x ðln xÞ2 ½K0 ðxÞ2 dx = 1 + ðγ − ln 2Þ + ðγ − ln 2Þ2 2

(A:2:106)

 ½Γ 41 4 ln x 2 pffiffiffi ½K0 ðxÞ dx = − 3=2 pffiffiffi ð3 ln 2 + γ + πÞ x 2 π

(A:2:107)

0 ∞ ð

0 ∞ ð

0 ∞ ð

1 x ðlnxÞ3 ½K0 ðxÞ2 dx = − ½6 + ðγ − ln 2Þ3 + 3 ðγ − ln 2Þ2 + 6 ðγ − ln 2Þ − ζ ð3Þ 2

0

(A:2:108) ∞ ð

x ln x K0 ðxÞK1 ðxÞ dx = −

π2 ð − 1 + 3 ln 2 + γÞ 8

(A:2:109)

x ðln xÞ2 K0 ðxÞK1 ðxÞ dx =

  π2 1 π2 ðγ + 3 ln 2Þ2 − ðγ + 3 ln 2Þ + 4 2 6

(A:2:110)

0 ∞ ð

0 ∞ ð

−∞

rffiffiffiffiffi     π 1 Bðα + , βÞ bα + β + 1=2 Kα + β + 1=2 ðajbjÞ jxj b − x Kν ðjaxjÞ Kμ ðajb − xjÞ dx = 2a 2 



α

Re α > −

β

1 2

;

Re β >

1 2

;

b − real (A:2:111)

∞ ð

0 ∞ ð

0





 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii π cosh a4b a 2 pffiffiffiffiffiffiffiffiffiffiffi J0 cosðbxÞK0 ðxÞ ber a ð1 + b Þ x dx = 2 4 2 1+b 2

2  2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i π sinh a4b a 2 p ffiffiffiffiffiffiffiffiffiffiffi sinðbxÞK0 ðxÞ bei a ð1 + b Þ x dt = J0 2 4 2 1+b

(A:2:112)

(A:2:113)

268

Appendix A The Bessel and Related Functions Integrals

ð1 x I0 ðaxÞK0 ðaxÞ Pn ð1 − 2x2 Þ dx = 0

1 ½In ðaÞ Kn ðaÞ + In + 1 ðaÞ Kn + 1 ðaÞ 2 ð2n + 1Þ (A:2:114)

ðτ

ðτ berðτ − xÞ beiðxÞ dx = 0

beiðτ − xÞ berðxÞ dx (A:2:115)

0

         1 τ τ τ τ = pffiffiffi sin pffiffiffi cosh pffiffiffi − cos pffiffiffi sinh pffiffiffi 2 2 2 2 2

A3 Integrals Containing Product of Three Bessel Functions and Elementary Functions ∞ ð

x Jλ ðaxÞ Jν ðbxÞ Kν − λ ðcxÞ dx =

α2 λ ðb2 − α2 Þ

ν−λ

2

aλ bν cν − λ ðβ − α2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α= ða + bÞ2 + c2 − ða − bÞ2 + c2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 β= ða + bÞ2 + c2 + ða − bÞ2 + c2 2 0

(A:3:1)

∞ ð

x2 Jλ − 1 ðaxÞ Jν ðbxÞ Kν − λ ðcxÞ dx = 0

"

ν−λ

2 α2 λ − 2 ðb2 − α2 Þ

aλ − 1 bν cν − λ ðβ2 − α2 Þ

2

λb2 − να2 −

# α2 ða2 + c2 − b2 Þ

(A:3:2)

β2 − α2

∞ ð

x2 Jλ + 1 ðaxÞ Jν ðbxÞ Kν − λ ðcxÞ dx = 0

" # ν−λ 2 α 2 λ − 2 ðb2 − α2 Þ b2 ðν b2 − λα2 Þ α2 ða2 + c2 − b2 Þ + 2 a λ − 1 bν c ν − λ ðβ2 − α2 Þ β2 ðβ2 − α2 Þ ðβ2 − α2 Þ

∞ ð

x J2ν ðaxÞ Iν ðb xÞ Kν ðb xÞ dx = 0 ∞ ð

0

1 a ða2

+ 4 b2 Þ1=2

;

 ν 23ν Γ ν + 21 ðab2 Þ xν + 1 Jν ðaxÞ Jν ðb xÞ Kν ðb xÞ dx = pffiffiffi π ða4 + 4 b4 Þ1=2

Re ν > −

(A:3:3)

1 2

; Re ν > −

(A:3:4) 1 2

(A:3:5)

Appendix A The Bessel and Related Functions Integrals

∞ ð

x I0 ðxÞ ½K0 ðxÞ2 dx =

π 33=2

0 ∞ ð

x3 I0 ðxÞ ½K0 ðxÞ2 dx =

269

(A:3:6)

4π 35=2

(A:3:7)

0 ∞ ð

  6 3 Γ 31 ½K0 ðxÞ dx = 2=3 2 · 32 π 3

0 ∞ ð

  1 6 Γ 28=3 π5 x ½K0 ðxÞ dx = 2=3 3 −   6 2 · 96 π 9 Γ 1 3

(A:3:10)

(   6 ) Γ 31 4 · 22=3 π5 x ½K0 ðxÞ dx = 2=3 −    2 · 96 π 9 Γ 1 6 3

(A:3:11)

2

0 ∞ ð

2

0 ∞ ð

(A:3:9)

3

3

x3 ½K0 ðxÞ3 dx = L − 3 ð2Þ − 0 ∞ ð

2 3

(A:3:12)

x6 ½K0 ðxÞ2 K1 ðxÞ dx =

− 162 + 147ζ ð3Þ 256

(A:3:13)

x8 ½K0 ðxÞ2 K1 ðxÞ dx =

− 37 + 63 ζ ð3Þ 8

(A:3:14)

0 ∞ ð

0

A4 Integrals Containing Product of Four and Five Bessel Functions and Elementary Functions ∞ ð

x Y0 ðxÞ K0 ðxÞ J1 ðα xÞ I1 ðα xÞ dx = − 0 ∞ ð

x I0 ðxÞ ½K0 ðxÞ3 dx =

π2 16

0 ∞ ð

lnð1 − α2 Þ 2π α2

;

0 > ∞ > > ð = < 1 ν+1 − Eið− xÞ = lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 x ξ

− Eið−xÞ = e − x lim

8 > >
>

Eið−xÞ = −

:

νν + 1

∞ ð

0

9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + xÞ ν2 + ξ 2 ξ + ν2 + ξ 2

8 > >
x > :

− Eið−xÞ = − e − x

∞ ð

0

(B:1:2) 9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + xÞ2 ν2 + ξ 2 ξ + ν2 + ξ 2

9 8 > > ∞ > > ð = < ðx + ξÞ −x ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ln x + e lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

− Eið−xÞ = e − x lim

8 > >
>

− Eið−xÞ = lim

(B:1:1)

8 > >
>

:

:

νν + 1

νν + 1

∞ ð

0

∞ ð

x

ξ x

ln qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

ν ! ∞> >

:

νν + 1

https://doi.org/10.1515/9783110681642-007

ðx

(B:1:5)

9 > > =



− Eið−xÞ = − lnðγxÞ − lim

(B:1:4)

9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðx + ξÞ ν2 + ξ 2 ξ + ν2 + ξ 2

8 > >
> =

ln qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2 0

(B:1:7)

280

Appendix B Limits Representing Special Functions

9 8 > > ∞ > > ð = < ln ξ −x ν+1 − Eið−xÞ = − e ln x + lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 x

(B:1:8)

8  h  i 9 ∞ > > ð tan − 1 ξ cos ν sin − 1 ξ = < x ν 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi − Eið−xÞ = dξ lim ν > π ν ! ∞> ; : ξ ν2 − ξ 2

(B:1:9)

0

− Ei½ − ða + xÞ = e − a lim

8 > >
>

− Eið−xÞ = 2e − x lim

8 > >
>

:

:

νν + 1

νν + 1

∞ ð

x

∞ ð

x

 ξ x

9 > > =

ln qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + aÞ ν2 + ξ 2 ξ + ν2 + ξ 2

(B:1:10)

9 >  pffiffiffiffiffiffi > = K0 2 x ξ dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:1:11)

8 h  i h  i 9 ∞ > > ð ln 1 + ξ sin ν sin − 1 ξ = < x ν 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi − Eið−xÞ = dξ ; x > 0 lim ν > π ν ! ∞> ; : ξ ν2 − ξ 2

(B:1:12)

0

− Eið− xÞ = e − x

( ∞ ð lim ν

ν!∞

0

e−νξ dξ x + ν sinh ξ

Ei½ − ðx + 1Þ + Eiðx − 1Þ = − 2 lim

8 > >
>

Ei½ − ðx + 1Þ − Eiðx − 1Þ = − 2 lim

:

8 > >
>

− Ei½ − ða + xÞ = e − a lim

8 > >
>

:

νν + 1

:

∞ ð

x

νν + 1

νν + 1

) (B:1:13) ∞ ð

1

∞ ð

1

9 > > =

coshðx ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ ν2 + ξ 2 ξ + ν 2 + ξ 2 9 > > =

sinhðx ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ ν2 + ξ 2 ξ + ν2 + ξ 2

9  > > = ln ξx qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + aÞ ν2 + ξ 2 ξ + ν2 + ξ 2

9 8  > > ∞ ξ > > ð = < ln x 2 −x ν+1 ½Eið− xÞ = 2 e lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 x ðx + ξÞ

(B:1:14)

(B:1:15)

(B:1:16)

(B:1:17)

Appendix B Limits Representing Special Functions

8  h  i 9 ∞ > > ð tan − 1 ξ cos ν sin − 1 ξ = < x ν 1 −x x qffiffiffiffiffiffiffiffiffiffiffiffiffi ½e EiðxÞ − e Eið−xÞ = lim ν dξ ν ! ∞> > 2 ; : ν2 − ξ 2 0

281

(B:1:18)

x>0

B.2 Sine and Cosine Integrals 9 8 > > ∞ > > ð = < π ½x cos x + ξ sin x ν+1 SiðxÞ = − lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q > 2 ν ! ∞> > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 ðx2 + ξ 2 Þ > 0 9 8 h  i > > = < ðx sin ν sin − 1 ξν qffiffiffiffiffiffiffiffiffiffiffiffiffi SiðxÞ = lim ν dξ ν ! ∞> > ; : ξ ν2 − ξ 2 0

(B:2:1)

(B:2:2)

9 8 h  i > > ð sin ν sin − 1 ξ = < ∞ ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi siðxÞ = − lim ν dξ ν ! ∞> > ; : ξ ν2 − ξ 2 x

(B:2:3)

9 8 h  i2 h  i ∞ −1 ξ > > ð ln x + ξ = < cos ν sin x−ξ ν 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi siðxÞ = − dξ lim ν > 2 π ν ! ∞> ; : ξ ν2 − ξ 2

(B:2:4)

0

CiðxÞ = lim

8 > >
>

:

νν + 1

∞ ð

0

9 > > =

½x sin x − ξ cos x dξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν > ; ν2 + ξ 2 ξ + ν2 + ξ 2 ðx2 + ξ 2 Þ >

9 8 h  i > > ð cos ν sin − 1 ξ = < ∞ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi CiðxÞ = − lim ν dξ ν ! ∞> > ; : ξ ν2 − ξ 2 x 8 > > >
ξ > > ln 1 + x = qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > > ν2 + ξ 2 ξ + ν2 + ξ 2 ;

1 (B:2:7) lim νν + 1 2 ν ! ∞> > > 0 : 9 8 > > ∞ > > ð = < 1 ν+1 cos x ciðxÞ − sin x siðxÞ = lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > 2 2 ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ðξ + x Þ cos x ciðxÞ − sin x siðxÞ =

(B:2:8)

282

Appendix B Limits Representing Special Functions

sin x ciðxÞ + cos x siðxÞ = − lim

8 > >
>

:

νν + 1



∞ ð

0

tan − 1 ξx

9 > > =

qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

sin x ciðxÞ + cos x siðxÞ = 9 8 > > ∞ > > ð = < 1 ν+1 − x lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > 2 2 ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ðξ + x Þ ( ∞ ð sin x ciðxÞ + cos x siðxÞ = − x lim ν ν!∞

0

(B:2:9)

(B:2:10)

)

e− νξ x2 + ðν sinh ξÞ2

dξ

( ∞ ) ð − νξ e ν sinh ξ dξ cos x ciðxÞ − sin x siðxÞ = lim ν ν!∞ x2 + ðν sinh ξÞ2

(B:2:11)

(B:2:12)

0

( ∞ pffiffiffi   ) ð − x2 =4 q e x π dξ sin x ciðxÞ + cos x siðxÞ = − lim ν pffiffiffi erfc ν ! ∞ q 2q 2 0

(B:2:13)

q = ν sinh ξ 9    2 > ξ > ∞ > ð ln 1 + x = 2 2 ν+1 ½siðxÞ + ½ciðxÞ = lim ν dξ   ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > > > ν2 + ξ 2 ξ + ν2 + ξ 2 0 ξ ; : 8 > > >
>
>

Γðx, αÞ = lim

:

8 > >
>

:

νν + 1

νν + 1

∞ ð

0

∞ ð

α

x

9 > > =

ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 x −1

(B:3:1)

9 > > =

ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > 2 ; ν2 + ξ ξ + ν 2 + ξ 2

9 8 > > x > > ð ξ=x = < e−αe x ν+1 Γð− x, αÞ = α x lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ; Re α > 0 q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

(B:3:2)

(B:3:3)

283

Appendix B Limits Representing Special Functions

Γðα, xÞ = e − x lim

8 > >
>

:

νν + 1

∞ ð

0

9 > > =

ðx + ξÞα − 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:3:4)

9 8 > > ∞ > > ð = < ξα −1 ν+1 Γðα, xÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 x

Γðα, xÞ =

8 > >
> :

Re α < 1; Γðα, xÞ = e

x

(B:3:5) 9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ ðξ − xÞα ν2 + ξ 2 ξ + ν2 + ξ 2

(B:3:6)

x>0 −x

( ∞ ð lim ν

ν!∞

0

− 1 < Re α < 1;

Γðα + 1, xÞ = e − x

)

e−νξ ðx + ν sinh ξÞ1 − α

dξ (B:3:7)

x>0 9 8 > > ∞ > > ð = < ξα ν+1 lim ν dξ   q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 x

(B:3:8)

( ∞   ð − ν ξ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ν x sinh ξ e Kα 1 2 e − x xα=2 Γ α, = dξ lim ν Γð1 − αÞ ν ! ∞ x ðν sinh ξÞα=2

(B:3:9)

0

− 1 < Re α < 1;

Γðα, − xÞ =

x>0 8 > >
> :

∞ ð

0

α

9 > > =

ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + xÞ ν2 + ξ 2 ξ + ν2 + ξ 2

(B:3:10)

Re α > − 1

Γð− α, xÞ =

8 > >
> :

∞ ð

x

9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ ðξ − xÞα ν2 + ξ 2 ξ + ν2 + ξ 2

Re α > − 1 (B:3:11)

284

Appendix B Limits Representing Special Functions

9 8 > > ∞ > > ð α = < 1 ðξ − xÞ ν+1 Γð− α, xÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi α > Γðα + 1Þ x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 x ξ

Γð− α, xÞ =

8 > >
> :

∞ ð

0

9 > p ffiffiffiffiffi ffi > = ξ α=2 Kα ð2 x ξ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

9 8 > > ∞ > > ð α −1 = < ξ ν+1   γðα, xÞ = lim ν dξ q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ΓðαÞ xα ν ! ∞> > > ξ =x Þ ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ð1 − e

(B:3:12)

(B:3:13)

(B:3:14)

Re α > 1

γðα, xÞ = lim

8 > >
> =

ξ νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ; > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

ν ! ∞> >

γðα, xÞ = lim

8 > >
>

:

νν + 1

∞ ð

x

9 > > = ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ >; > ; ν2 + ξ 2 ξ + ν 2 + ξ 2 α −1

Re α > 0

(B:3:15)

Re α > 0

(B:3:16)

9 8 > > x > > ð − ξ=x = < αx e−αe ν+1   ; Re α > 0 γðx, αÞ = dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > x ν ! ∞> > > 2 2 2 2 ; : ν +ξ ξ + ν +ξ 0

γðα, xÞ = xα=2 lim

8 > >
>

:

νν + 1

γðα, xÞ = ΓðαÞ xα=2 e − x

∞ ð

0

9 > p ffiffiffiffiffi ffi > = ξ α=2 − 1 Jα ð2 xξ Þ dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν >; > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

Re α > 0

9 8 > > ∞ p ffiffiffiffiffi ffi > > ð = < Iα ð2 x ξ Þ ν+1 lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > α=2 ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ξ

9 8 > > > > ðx α −1 = < ðx − ξÞ x ν+1 γðα, − xÞ = e lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ; Re α > 0 q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

(B:3:17)

(B:3:18)

(B:3:19)

(B:3:20)

Appendix B Limits Representing Special Functions

9 8 > > ∞  > > ð y − 1 = < ξ=x 1 − e 1 ν+1 Bðx, yÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi > x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

285

(B:3:21)

Re y > 0

ψðxÞ = ln x + lim

8 > >
>

ψðxÞ = − lim

8 > >
>

:

:

νν + 1

νν + 1

∞ ð

0

∞ ð

0

9 > > =

eξ=x qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðeξ=x − 1Þ ν2 + ξ 2 ξ + ν2 + ξ 2

9 >   > = ln γ ðeξ=x − 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:3:23)

9 8 h  i > > ∞ ξ > > 2x ð = < − csch ξ 2x 1 ν+1 ψðxÞ = −  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ + ln x + 2x lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > 2x > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 9 8    μ−1 > > ξ ξ > > ∞ > > ð ctnh 2 x = < 2x ΓðμÞ μ ν+1 ψðxÞ = + 2 x lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > 2 xμ > > > > ν2 + ξ 2 ξ + ν2 + ξ 2 0 ; :

8 h h  i  i 9 > > ∞ ξ > > ð = < ln sinh 2 x − ln 2ξx 1 1 ν+1   ψðxÞ = dξ + lnð2 xÞ − lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 4x 2 x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 x 1 x ψ = − ln + x lim νν + 1 ν ! ∞> 2 x 2 > : 8 > >
> :

∞ ð

0

∞ ð

0

n

h

x ξ ln

 io ξ x

 i ξ x

(B:3:27)

(B:3:28)

9 > > =

− cosh qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 x ξ

(B:3:26)

9 > > =

sinh qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 h

(B:3:24)

(B:3:25)

9 8 h  i > > ∞ ξ > > 2x ð = < ξ − ctnh 2 x 1 ν+1 ψðxÞ = −  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ + ln x + 2x lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > 2x > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

8 > >
>
> :

∞ ð

0

h

 i ξ x

9 > > =

− cosh qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 x ξ

(B:3:30)

9 8 > > ∞ > >   ð = < x 2 x+1 1 ν+1 ψ −ψ  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ = lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2 x ν ! ∞> > > − ξ =x Þ ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ð1 + e (B:3:31) 8 > >
> =

 − ξ=x ∞   1+e ð  ln 2 x+1 x ψ −ψ = 2x lim νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ν ! ∞> > 2 2 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 9 8 h  i > > ∞ ξ > >   x ð = < x 1 − csc h x ξ x+1 ψ = ln + lim νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > 2 2 x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

ψðnÞ ðxÞ =

 −

1 x

n + 1

(B:3:32)

(B:3:33)

9 8 > > ∞ > > ð = < ξn ν+1 lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > − ξ=x − 1Þ ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ðe

n = 1, 2, 3, . . . (B:3:34)

B.4 Error Functions and Fresnel Integrals 9 8 > > ∞ pffiffiffi > > ð = < sinð2x ξ Þ 1 ν+1 erfðxÞ =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > π ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ξ 8 h  i 9 ∞ > > ð e − ξ 2 =4 x2 sin ν sin − 1 ξ = < ν 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffi erfðxÞ = dξ lim ν > π ν ! ∞> ; : ξ ν2 − ξ 2

(B:4:1)

(B:4:2)

0

9 8 > > p ffiffiffiffiffi ffi ∞   > > ð = < pffiffiffi coshð2 xξ Þ − 1 e−x ν+1 p ffiffi ffi erfð xÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q > π x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:4:3)

Appendix B Limits Representing Special Functions

9 8 > > ∞ > > ð = < e 1 ν+1 p ffiffiffi erfcðxÞ =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffiffiffiffiffiffiffiffi > π ν ! ∞> > > ; : ξ + x2 ν2 + ξ 2 ξ + ν2 + ξ 2 0

287

x2

9 qffiffiffiffiffiffiffiffiffiffiffiffiffi  > 2 > ∞ > ð x2 + ξ − x 2 = 2e − x ν+1 erfcðxÞ = pffiffiffi lim ν dξ   ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > π ν ! ∞> > > > > ν2 + ξ 2 ξ + ν2 + ξ 2 0 ; :

(B:4:4)

8 > > >
>
> : 8 > >
> : 2

∞ ð

0

∞ ð

0

− 2 x2

(B:4:5) 9 > > =

pffiffi ξ

e  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi2 > > ; ξ ν2 + ξ ξ + ν 2 + ξ 2 9 > pffiffiffi > = sinhð2x ξ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

8 > >
> :

∞ ð

0

(B:4:7)

93 > > =7 erfc 7 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ >7 5 > 2 2 2 2 ; ν +ξ ξ + ν +ξ  ξ 2x

9 8 > > ∞ > > pffiffiffi − x ð = < pffiffiffi 1 xe ν+1 erfcð xÞ = dξ   lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > pffiffiffi ν ! ∞> > π > ; : ξ ðξ + xÞ ν2 + ξ 2 ξ + ν2 + ξ 2 0 8 > >
> :

∞ ð

0

0

pffiffiffi erfcð xÞ = e − x 1 − 2 lim

e−2 xξ  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > > 2 2 ; ξ ν + ξ ξ + ν2 + ξ 2

ν!∞

CðxÞ =

8 > >
> :

0 ∞ ð

0

(B:4:8)

(B:4:9)

9 > > =

pffiffiffiffi

( ∞ ) ð pffiffiffi e − x e−νξ erfcð xÞ = pffiffiffi lim ν pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dξ π ν!∞ x + ν sinh ξ "

(B:4:6)

(B:4:10)

(B:4:11) )#

e−νξ ðx + ν sinh ξÞ3=2 

ξ2 4x

dξ

(B:4:12) 9 > > =

sin −x qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 + ν2 ξ + ξ 2 + ν2

(B:4:13)

288

Appendix B Limits Representing Special Functions

SðxÞ = 

8 > >
> :

2 1 − SðxÞ + 2 8 > > > < 2 lim νν + 1 π ν ! ∞> > > :



1 − CðxÞ 2

∞ ð

0

∞ ð

0

9 > > =

ξ2

cosð4x − xÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 + ν2 ξ + ξ 2 + ν2

(B:4:14)

9 > > > =

(B:4:15)

2 =   2 ξ 2x

sin qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > > ξ ξ 2 + ν2 ξ + ξ 2 + ν2 ;



 2 2   2 2 1 x 1 x − = −C −S 4 4 2 2 9 8 > > ∞ > > ð = < 2 1 Siðxξ Þ ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q > π ν ! ∞> > > ; : ξ 2 + ν2 ξ + ξ 2 + ν2 0     π π sin x sin x + CðxÞ − cos x + SðxÞ = pffiffiffiffiffiffi 4 4 2π s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( ∞ ) ð 1 ν sinh ξ − νξ + pffiffiffi lim ν e 1− dξ 2 π ν!∞ x2 + ðν sinh ξÞ2

(B:4:16)

(B:4:17)

0

    π π sin x 1 CðxÞ  cos x þ SðxÞ ¼ pffiffiffi  pffiffiffiffiffiffi  sin x þ 4 4 2 πx 2 9 8 ∞ >   >  ð = < νξ 1 e 3 x dξ sin lim ν h tan 1 i 3=2 3=4 > 2 ν sinh ξ 2 π ν ! ∞> 2 ; : 2 0 x þ ðν sinh ξÞ

(B:4:18)

B.5 Legendre Functions 9 8 > > ∞ > >   ð λ = < ξ I − μ ðx ξÞ 1 1 μ ν+1 Pλ pffiffiffiffiffiffiffiffiffiffiffi =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q > > Γðλ − μ + 1Þ ν ! ∞> 1 − x2 > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:5:1)

289

Appendix B Limits Representing Special Functions

Pμ− λ ðxÞ =

8 > >
> :

∞ ð

0

μ

9 > > =



Iλ ξx

ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

(B:5:2)

Reðλ + μÞ − 1 9 8  > > ∞ ξ μ − 1=2 > > ð = < ξ Kλ + 1=2 x −μ ν+1 Pλ ðxÞ = dξ   lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > Γðλ + μ + 1Þ Γðμ − λÞ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 qffiffiffiffiffi 2 πx

Reðλ + μÞ − 1; μ

Qλ ðxÞ =

pffiffiffiffi π

2x

(B:5:3)

Reðμ − λÞ > 0 8 > >
sinðπ λÞ > :

∞ ð

0

μ − 1=2



Iλ + 1=2 ξx

9 > > =

ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:5:4)

Reðλ + μÞ − 1 9 8  > > ∞ ξ μ > > ð = < ξ Kλ x sin½πðλ + μÞ μ ν+1 Qλ ðxÞ = dξ   lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > Γðλ + μ + 1Þ sinðπ λÞ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:5:5)

Reðμ ± λÞ − 1

B.6 Hypergeometric and Confluent Hypergeometric Functions 2 F1 ðμ, α;α + β + 1; xÞ =

8 > >
> : β > −1,

j xj < 1

2 F1 ðα, α +

1 ; β; x2 Þ = 2

2

2α−β

ΓðαÞ x Γð2 αÞ

Re α > 0;

1 −α

8 > >
>

Re β > 0

:

∞ ð

0

9 > > =

− ξ =α β

ð1 − e Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > μ > ; ν2 + ξ 2 ξ + ν 2 + ξ 2 ð1 − x e − ξ=α Þ

νν + 1

∞ ð

0

2α −β

9 > > =

ξ Iβ − 1 ðx ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

(B:6:1)

(B:6:2)

290

Appendix B Limits Representing Special Functions

1 ; β; − x2 Þ = 2 9 8 > > ∞ > > ð 2 α − β = < ξ Jβ − 1 ðxξÞ 22 α − β ΓðαÞ x1 − α ν+1 dξ   lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ν ! ∞> Γð2 αÞ > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

2 F1 ðα, α +

Re α > 0;

Re β > 0

2 F1 ðα, α + 1; 2α; − x

8 > >
> : Re α > −

(B:6:3)

∞ ð

0

9 h  i2 > xξ > = ξ Jα − 1=2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:6:4)

1 2

  3 3 1 − μ, − 2 μ; 2 − μ; − 2 = 2 2 4x 9 8 h  i ∞ > > ð ξ 1 − 2 μ Kμ ðx ξÞ2 sin ν sin − 1 ξ = < ν 8 Γð2 − μÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi q ν lim dξ > π Γð32 − μÞ Γð32 − 2μÞ ν ! ∞> ; : ν2 − ξ 2

2 F1

(B:6:5)

0

Re μ
> ∞  xξ > > ð ξ2α −1 J 1 −2α = < 1 α − 1=2 2 π Γ α + 2 ð2xÞ ν+1   ν dξ lim q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ΓðαÞ Γð2 α − 21Þ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 Re α >

(B:6:6)

1 2

1 F1 ðλ + μ;2λ + 1;

− xÞ =

9 8 > > ∞ p ffiffiffiffiffiffiffi ffi > > p ffiffi ffi ð = < ξ μ − 1 J2 λ ð2 x ξÞ Γð2λ + 1Þ ex=2 x ν+1 lim ν dξ   q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > Γðλ + μÞ xλ κ − μ + 21 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ðx − ξÞ

Reðλ + μÞ > 0 (B:6:7)

Appendix B Limits Representing Special Functions

 1 3 1 − μ, − 2μ; 2 − μ; − 2 = 1 F1 2 2 4x 9 8  i h ∞ > > ð ξ 2 μ e − x ξ 2 Iμ ðxξ 2 Þ sin ν sin − 1 ξ = < 2 μ − 3=2 1=8 x ν 2 Γð1 − μÞ e ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffi ffi q ν lim dξ 1 ν ! ∞> > Γðμ + 2Þ x ; : ν2 − ξ 2

291



(B:6:8)

0

Rejμj
> ∞ > > ð = < π πk I0 ðk ξÞI1 ðk ξÞ ν+1 EðkÞ = − dξ   lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 2 2 ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ξ

EðkÞ =

8 > >
> :

∞ ð

0

9 h  i2 > kξ > = I0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eð2k 1 + kÞ = π ð1 − kÞ lim

8 > >
>

EðkÞ =

8 > >
2 > :

∞ ð

0

:

νν + 1

∞ ð

0

9 > pffiffiffi > = ξ J0 ðk ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

 h   i 9 > kξ kξ > = I0 2 I0 2 + k ξ I1 k2ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

9 8   > > ∞ kξ kξ > > ð = < I0 2 K0 2 pffiffiffiffiffiffiffiffiffiffiffi ν + 1 Kð 1 − k2 Þ = lim ν dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ν! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 9 8 h  i2 > > pffiffiffiffiffiffiffiffiffiffiffi ∞ kξ > >   ð = < I0 2 k π 1 + k2 ν+1 K pffiffiffiffiffiffiffiffiffiffiffi = dξ   lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν! ∞> > 2 1 + k2 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 9 8   > > pffiffiffiffiffiffiffiffiffiffiffi ∞ kξ kξ > >   ð = < J0 2 Y0 2 1 π 1 + k2 ν+1 K pffiffiffiffiffiffiffiffiffiffiffi = −  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν! ∞> > 2 1 + k2 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

(B:7:1)

(B:7:2)

(B:7:3)

(B:7:4)

(B:7:5)

(B:7:6)

(B:7:7)

292

Appendix B Limits Representing Special Functions

9 8 h  i2 > > pffiffiffiffiffiffiffiffiffiffiffi ∞ kξ > >   ð = < J0 2 k π 1 + k2 ν+1 ffiffiffiffiffiffiffiffiffiffiffi p = E  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν! ∞> > 2 1 + k2 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:7:8)

9 8 > > ∞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ð = < pffiffiffiffiffiffiffiffiffi π ð1 + kÞ I0 ðk ξÞ ν+1 Kð2 k 1 + kÞ =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν! ∞> > 2 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ξ 9 8 h  i2 h  i2 > > ∞ kξ > > ð I0 k ξ = < + I 1 2 2 πk ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ EðkÞ − KðkÞ = lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > 4 ν! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 2

EðkÞ − ð1 − k2 Þ KðkÞ = 9 8   > > ∞ kξ kξ > > ð = < 2 ξ I I 0 2 1 2 π k ð1 − k Þ ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν! ∞> > 2 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:7:9)

(B:7:10)

9 8 h  i2 > > ∞ kξ > > ð = < I1 2 π k2 2 ν+1 ð2 − k Þ KðkÞ − 2EðkÞ =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 ν! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0     1 k k pffiffiffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffiffiffi − E pffiffiffiffiffiffiffiffiffiffiffi = 1 + k2 1 + k2 1 + k2 9 8   > > ∞ kξ kξ > > ð = < ξ J J 0 2 1 2 πk ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 ν! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:7:11)

(B:7:12)

B.8 Parabolic Cylinder and Whittaker Functions − x2 =4

8 > >
> : Re μ < 1

∞ ð

0

μ=2

9 > > =

ξ − ðμ + 1Þ=2 ðx2 + 2 ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:8:1)

Appendix B Limits Representing Special Functions

Dμ ðxÞ =

8 > >
>

:

νν + 1

9 > p ffiffiffiffiffi ffi  > = ξ ðμ − 1Þ=2 cos π2μ − x 2 ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

∞ ð

0

293

(B:8:2)

Re μ > − 1

D − μ ðxÞ =

8 > >
> : 8 > >
> :

Re μ > 0 pffiffiffi D − 2 μ ð xÞ = − x2 =2

∞ ð

8 > >
> :

∞ ð

x

9 > > =

ξμ −1 e qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ >; > ; ν2 + ξ 2 ξ + ν2 + ξ 2 − ξ 2 =2 x2

∞ ð

0

pffiffi 2

Re μ > 0

(B:8:3)

9 > > =

ξμ −1 e−x ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:8:4)

9 > > =

ξμ −1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + xÞμ + 1=2 ν2 + ξ 2 ξ + ν2 + ξ 2

(B:8:5)

Re μ > 0 pffiffiffi D − 2 μ ð2 xÞ =

9 8 > > ∞ > > p ffiffi ffi ð = < μ − 1 e − x= x ξ ν+1 ν dξ   lim qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > > 2μ − 1 ΓðμÞ ν ! ∞> μ + 1=2 > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ðξ + xÞ

(B:8:6)

Re μ > 0 pffiffiffi D − 2 μ ð2 xÞ =

9 8 > > ∞ > > pffiffiffi ð μ −1 = < ðξ − xÞ x ν+1 ν dξ   lim q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 2μ − 1=2 ΓðμÞ ν ! ∞> μ + 1=2 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 x ðξ + xÞ

Re μ > 0

(B:8:7)

294

Appendix B Limits Representing Special Functions

pffiffiffi D − 2 μ ð xÞ =

8 > >
> :

∞ ð

0

pffiffiffiffiffiffi

9 > > =

ξμ −1 e− 2xξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

(B:8:8)

Re μ > 0

D − 2 μ ðxÞ =

x> −

2

8 > >
>

:

νν + 1

∞ ð

0

9  2 > > = γ μ, 8ξx2 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

(B:8:9)

1 2

9 8 > > ∞ pffiffiffiffiffiffiffiffi > > ð μ − 1 = < pffiffiffi 2μ + 1=2 e x=4 ξ sinðπ μ − 2x ξ Þ ν+1 pffiffiffi dξ   lim ν D2 μ − 1 ð xÞ = qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ν ! ∞> π > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:8:10)

Re μ > 0 pffiffiffi D1 − 2 μ ð2 xÞ = 1 2μ − 1=2

ΓðμÞ

8 > >
>

:

νν + 1

∞ ð

x

9 > > =

ðξ − xÞμ − 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + xÞμ − 1=2 ν2 + ξ 2 ξ + ν2 + ξ 2

(B:8:11)

Re μ > 0 pffiffiffi D1 − 2 μ ð2 xÞ =

8 > >
> : Re μ > 0

∞ ð

0

9 > > =

ξμ −1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ðξ + xÞμ − 1=2 ν2 + ξ 2 ξ + ν2 + ξ 2

(B:8:12)

Appendix B Limits Representing Special Functions

x 1 − μ ex=2 I Mκ, μ ðxÞ =  B κ + μ + 21 , κ − μ + 21 9 8 > > ∞ > > 1 ð = < ξκ+μ+2 ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ I = lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν!∞ > > κ − μ + 21 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ðx − ξÞ 1 μ±κ> − ; 2

295

(B:8:13)

x>0

Mκ, μ ðxÞ =

9 8 > > ∞ pffiffiffiffiffiffi > > pffiffiffi − x=2 ð = < I2 μ ð2 xξ Þ Γð2μ + 1Þ x e ν+1 ν dξ   lim q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > Γðμ − κ + 21Þ > > κ + 21 ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ξ

Reðκ − μÞ
> ∞ pffiffiffiffiffiffiffiffi > > p ffiffi ffi 1 ð κ − = < ξ 2 Jμ ð2 xξÞ Γð2μ + 1Þ ex=2 x ν + 1  ν dξ   lim q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > Γ μ + κ + 21 κ − μ + 21 > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ðx − ξÞ

Reðμ + κÞ > −

1 2 (B:8:15)

M ðμ, − 1Þ=2, μ=2 ðxÞ =

x > 0;

e

− ðμ − 1Þ=2

μ

9 8 > > ∞ > > ð = < xe−ξ −ξ μ −1 e ð1 − e Þ ν+1  ν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:8:16)

Re μ > 0

9 8 > > ∞ > > 1 1 ð μ + κ − = < 2 e − α=2 x1=2 − μ ξ μ − κ − 2 ðx + ξÞ ν + 1 Wκ, μ ðxÞ =  ν dξ   lim qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 1 ν ! ∞> Γ μ−κ+ 2 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0   1 Re μ − κ + >0 2

(B:8:17)

296

Appendix B Limits Representing Special Functions

W − μ, λ ðxÞ =

9 8 > > ∞ p ffiffiffiffiffi ffi > > pffiffiffi ð = < ξ μ − 1=2 K2 λ ð2 x ξ Þ 2 x ν + 1   ν dξ   lim qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 1 1 ν ! ∞> Γ λ+μ+ 2 Γ μ−λ+ 2 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:8:18)

Re λ > − 1 9 8  μ =2  > > ∞ μ ξ 2ξ > > ð = < 1+ x Pκ 1 + x e − x=2 ν+1 Wμ, κ + 1=2 ðxÞ = dξ   lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:8:19)

Re μ < 1

Wμ, κ + 1=2 ð2 xÞ = lim

8 > >
>

:

νν + 1

∞ ð

x



ξ +x ξ −x

μ =2

μ Pκ

 ξ x

9 > > =

qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

Re μ > 1;

W − μ,

x > 0;

κ≠0, ± 1, ± 2, ± 3, ... 9 8 > > ∞ > > ð − ξ μ − 1 = < ex=2 e − x e ð1 − e − ξ Þ ν+1 ðxÞ = ν dξ   lim q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi − 1=2 ν > ΓðμÞ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

(B:8:20)

(B:8:21)

Re μ > 0

B.9 Legendre, Laguerre and Hermite Orthogonal Polynomials 9 8  > > ∞ > > ð 2 F2 − n, n + 1; 1, 1; x ξ = < 2 ν+1 Pn ð1 − xÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 9 8 > > ∞ > >   ð ðn − 1Þ=2 = < 1 ð1 + x2 Þ ξ n J0 ðx ξÞ ν+1 Pn pffiffiffiffiffiffiffiffiffiffiffi = dξ   lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > n! 1 + x2 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 9 8 > > ∞ > >   ð ðn + 1Þ=2 = < 1 ð1 − x2 Þ ξ n I0 ðx ξÞ ν+1 Pn pffiffiffiffiffiffiffiffiffiffiffi = dξ   lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > n! 1 − x2 > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:9:1)

(B:9:2)

(B:9:3)

297

Appendix B Limits Representing Special Functions

9 8 > > ∞ p ffiffiffiffiffi ffi > > ð n + α=2 = < ξ J ð2 x ξ Þ 1 α α ν+1 Ln ðxÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q α=2 > n! x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:9:4)

Reðα + nÞ > − 1 8 > >
> :

∞ ð

0

9 pffiffiffi > π n > = ξ cos 2 − 2 x ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 ðn − 1Þ=2

(B:9:5)

n = 0, 1, 2, 3, ... 8 > >
π > :

∞ ð

0

9 > p ffiffiffiffiffi ffi > = ξ n − 1=2 cosð2 x ξ Þ dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:9:6)

n = 0, 1, 2, 3, ... 9 8 > > ∞ pffiffiffiffiffiffi > > ð n − 1=2 = < pffiffiffiffiffi ð− 1Þn 2n + 1=2 ex ξ sinð2 x ξ Þ ν+1 pffiffiffi He2 n + 1 2 x =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q ν ! ∞> > π > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 n = 0, 1, 2, 3, ... (B:9:7)

He2 n ðxÞ =

n 3=2 1 − 2 n x2

ð− 1Þ 2

x pffiffiffi π

e

8 h  i 9 > > ð ξ 2 n e − ξ 2 =4 x2 cos ν sin − 1 ξ = < ∞ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi lim ν dξ ν ! ∞> > ; : ν2 − ξ 2 0

(B:9:8)

B.10 Riemann Zeta Functions ζ ðz − 1Þ =

z

8 > >
> :

∞ ð

0

9 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii > > = tanh π2 e2ξ=z − 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:10:1)

298

Appendix B Limits Representing Special Functions

ζ ðμ, xÞ =

8 > >
> :

Re μ > 0, ζ ðμ, xÞ =

νν +

∞ ð 1 0

9 > > =

ξμ −1 dξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν > 2 2 2 2 ξ=ðx − 1Þ ; ν + ξ ξ + ν + ξ ðe − 1Þ >

(B:10:2)

x>1 8 > >
> :

∞ ð 1 0

9  > ξ > = ξ csch 2 x − 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ð2x − 1Þμ − 1 ν2 + ξ 2 ξ + ν2 + ξ 2 μ −1

(B:10:3)

Re μ > 1 ζ ðμ, xÞ =

9 8 h  i > > ∞ ξ μ −1 > > ð = < ξ ctnh 2 x − 1 − 1 1 ν+ 1 ν dξ   lim q ffiffiffiffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffiffiffiffi ffi ν > 22 μ − 1 ΓðμÞ ðx − 1Þ ν ! ∞> μ −1 > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ð2 x − 1Þ

(B:10:4)

Re μ > 1 8 > >   < 1+x ð2 xÞμ ζ μ, = lim νν + 2 ν ! ∞> 2 > :

∞ ð 1 0

9  > ξ > = ξ csch x qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 μ −1

(B:10:5)

Re μ > 1

B.11 Volterra Functions

( ∞    ) ð 1 1 x=ξ dξ νðxÞ = e − lim λ Jλ ðλx ξÞ e − ν λ! ∞ ξ ξ x

8 > >
> : 8 > >
> :

0

9 >  pffiffiffiffiffiffi > = ν 2 xξ  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi > > ; ξ ν2 + ξ 2 ξ + ν2 + ξ 2 ∞ ð

1 0

9 >  pffiffiffiffiffiffi > = ν 2 xξ, 2α  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffi > > ; ξ ν2 + ξ 2 ξ + ν2 + ξ 2

(B11:1)

(B11:2)

(B11:3)

Appendix B Limits Representing Special Functions

8 > >
> : ( ∞ ð νðx, αÞ = x lim λ α

λ! ∞

0

νðe

−x

, αÞ = e

−αx

∞ ð 1 0

9 >  pffiffiffiffiffiffi > = ν 2 xξ, 1 + 2α qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν2 + ξ 2

) 1 1=ξ Jλ ðλξÞ νðx , αÞ dξ ; ξ xα=ξ

Re α > − 1

( ∞ ) ð 1 α x=ξ − x=ξ lim λ νðe , αÞ dξ Jλ ðλ ξÞ e λ! ∞ ξ

299

(B11:4)

(B11:5)

(B11:6)

0

( ∞ ) ð x ð1 − ξÞ Jλ ðλξÞ νðα, x ξÞ dξ ; νðα, xÞ = lim λ α λ! ∞

0

μðx, αÞ =

1 2α+1

8 > >
> :

∞ ð 1 0

x> −1

9 > pffiffiffiffiffiffi > = μð2 xξ , αÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffi > > ; ξ ν2 + ξ 2 ξ + ν2 + ξ 2

(B11:7)

(B11:8)

B.12 Bessel and Related Functions 8 h  i  9 ∞ > > ð sin ν sin − 1 ξ sin x2 = < 4ξ ν 4 qffiffiffiffiffiffiffiffiffiffiffiffiffi J0 ðxÞ = dξ lim ν > π x ν ! ∞> ; : ν2 − ξ 2

(B:12:1)

0

8 h  i  9 ∞ > > ð sin ν sin − 1 ξ sin x2 = < 4ξ ν 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi Y0 ðxÞ = − K0 ðxÞ + dξ lim ν > π π ν ! ∞> ; : ξ ν2 − ξ 2

(B:12:2)

0

8 h  i  9 ∞ > > ð sin ν sin − 1 ξ sin x2 = < 4ξ ν x qffiffiffiffiffiffiffiffiffiffiffiffiffi J1 ðxÞ = dξ lim ν > π ν ! ∞> ; : ξ 2 ν2 − ξ 2 0 sin x J0 ðxÞ − cos x Y0 ðxÞ = 9 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > 2 ∞ pffiffiffi > > 2 ð = < ξ + ξ + 4x 2 ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q > π ν ! ∞> > > > > ξ ðξ 2 + 4x2 Þ ν2 + ξ 2 ξ + ν2 + ξ 2 0 ; :

(B:12:3)

(B:12:4)

300

Appendix B Limits Representing Special Functions

cos x J0 ðxÞ + sin x Y0 ðxÞ = 9 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > > 2 ∞ pffiffiffi > > ð = < ξ + 4x2 − ξ 2 ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q > π ν ! ∞> > > > > ξ ðξ 2 + 4x2 Þ ν2 + ξ 2 ξ + ν2 + ξ 2 0 ; :

(B:12:5)

sin x J1 ðxÞ − cos x Y1 ðxÞ = 9 8  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2 > > 2 > > 2 ∞ > > ð ξ + ξ + 4x = < 1 pffiffiffi lim νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ  > π 2 x ν ! ∞> > > > > ν2 + ξ 2 ξ + ν 2 + ξ 2 ξ ξ 2 + 4x2 0 ; :

(B:12:6)

cos x J1 ðxÞ + sin x Y1 ðxÞ = 9 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 > > 2 > > 2 ∞ > > ð ξ + 4x − ξ = < 1 ν+1 − pffiffiffi lim ν dξ   ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν  ffi > π 2 x ν ! ∞> > > > > ν2 + ξ 2 ξ + ν 2 + ξ 2 ξ ξ 2 + 4x2 0 ; :

(B:12:7)

J02 ðxÞ + Y02 ðxÞ =

8 > > >
> > :

∞ ð

0

9  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > K ξ ξ 2 + 4x2 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > > ξ 2 + 4x2 ν2 + ξ 2 ξ + ν2 + ξ 2 ;

(B:12:8)

9 8 h  i > ðx cos ν sin − 1 ξ x2 − ξ 2 μ − 1=2 > = < ν 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffi Jμ ðxÞ = μ − 1 pffiffiffi  lim ν dξ > 2 π Γ μ + 21 xμ ν ! ∞ > ; : ν2 − ξ 2 0

Re μ > −

1 2 (B:12:9)

1 I Jμ ðxÞ = pffiffiffi  1 π Γ μ + 2 ð2 xÞμ cos x 8 h  i μ − 1=2 9 > > = < 2ðx cos ν sin − 1 ξν 2 xξ − ξ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi I = lim ν dξ ν!∞ > > ; : ν2 − ξ 2 0 Re μ > −

1 2

(B:12:10)

301

Appendix B Limits Representing Special Functions

1 I pffiffiffi  2μ − 1 π Γ μ + 21 9 8 h  i μ − 1=2 > > = < ðx sin ν sin − 1 ξν ðx2 − ξ 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ I = lim ν ν ! ∞> > ; : ν2 − ξ 2

Jμ ðxÞ =

(B:12:11)

0

Re μ > −

1 2

1 Jμ ðxÞ = pffiffiffi  I π Γ μ + 21 ð2 xÞμ sin x 9 8 h  i μ − 1=2 > > = < ðx sin ν sin − 1 ξν ð2 xξ − ξ 2 Þ q ffiffiffiffiffiffiffiffiffiffiffiffiffi I = lim ν dξ ν ! ∞> > ; : ν2 − ξ 2 0 Re μ > −

(B:12:12)

1 2

1 pffiffiffi I 2μ − 1 π Γðμ + 21Þ xμ + 1 9 8 h  i μ − 1=2 > > = < ðx sin ν sin − 1 ξν ξ ðx2 − ξ 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi I = lim ν dξ ν ! ∞> > ; : ν2 − ξ 2 0 Jμ + 1 ðxÞ =

Re μ > −

(B:12:13)

1 2

9 8 h  i > ð sin ν sin − 1 ξ ðξ 2 − x2 Þμ − 1=2 > = < ∞ ν 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffi lim ν J − μ ðxÞ = μ − 1 pffiffiffi dξ > > 2 π Γðμ + 21Þ xμ ν ! ∞: ; ν2 − ξ 2 x

jRe μj
> ð = < cos ν sin ν 2μ + 1 x μ pffiffiffi lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ Yμ ðxÞ = − 1 μ + 1=2 > Γð2 − μÞ π ν ! ∞ > 2 2 : ν2 − ξ 2 ; x ðξ − x Þ jRe μj
> ð sin ν sin − 1 ξ ξ ðξ 2 − x2 Þμ − 1=2 = < ∞ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi lim ν dξ ν!∞ > > ; : ν2 − ξ 2 x

1 < Re μ < 0 2 (B:12:16) 8 > >
> :

I − 5=4 ðxÞ =

8 > > >
> > :

9 > > =

 2 2 e − x =2 ξ Kμ 2xξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ ν2 + ξ 2 ξ + ν2 + ξ 2



3 8ξ

0

(B:12:17)

pffiffiffiffiffiffiffi  9 ffi coshpffiffiffiffiffiffi > 8xξ 3 sinh 8xξ > > −1 − = ξ 3=4 ξ 5=4 dξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν > > > ν2 + ξ 2 ξ + ν2 + ξ 2 ;

(B:12:18) 9 8 > > p ffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffi ffi ∞     p ffiffi ffi > > ð = < 23=2 x sinh 8x ξ − cosh 8x ξ e−x ν+1 I − 3=4 ðxÞ = 7=4 pffiffiffi lim ν dξ   q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 2 π x ν ! ∞> > > ; : ξ 3=4 ν2 + ξ 2 ξ + ν2 + ξ 2 0 (B:12:19)

I − 1=4 ðxÞ =

8 > >
>

:

ν+1

∞ ð

ν

0

9 > > =

pffiffiffiffiffiffiffiffiffi cosh 8x ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 3=4 ν2 + ξ 2 ξ + ν2 + ξ 2

8 h  i 9 ∞ > > ð e − 1=8 x ξ cos ν sin − 1 ξ = < ν e ffiffiffiffiffiffiffiffiffiffiffiffiffi q I − 1=4 ðxÞ = pffiffiffi lim ν dξ p ffiffi ffi > π x ν ! ∞> ; : ξ ν2 − ξ 2

(B:12:20)

x

(B:12:21)

0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 h  i > > ∞ ξ − 1 pffiffiffi 2 x > = < ð sin ν sin ξ + ξ 2 + 4x2 > ν 2e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi I0 ðxÞ = dξ lim ν > π ν ! ∞> > > ξ ðξ 2 + 4x2 Þ ν2 − ξ 2 ; : 0

I1=4 ðxÞ =

e−x π ð2xÞ1=4

9 8 > > ∞ p ffiffiffiffiffiffiffiffi ffi > > ð = < sinhð 8x ξ Þ lim νν + 1 dξ   q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > 3=4 ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ξ

(B:12:22)

(B:12:23)

303

Appendix B Limits Representing Special Functions

9 8 > > p ffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffi ffi ∞   p ffiffi ffi > > ð = < 3=2 2 x coshð 8x ξ Þ − sinhð 8xξ Þ e−x ν+1 p ffiffi ffi I3=4 ðxÞ = 7=4 dξ lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 π x ν ! ∞> > > ; : ξ 3=4 ν2 + ξ 2 ξ + ν2 + ξ 2 0 (B:12:24)

I5=4 ðxÞ =

I0 ðxÞ =

I1 ðxÞ =

8 > > >
> > : 8 > >
> : 8 > >
> :

2ðx

0

2ðx

0

∞ ð

0



pffiffiffiffiffiffiffi

3 8ξ

9 > > > =

pffiffiffiffiffiffiffi 

sinh 8 x ξ 3 cosh 8xξ +1 − ξ 3=4 ξ 5=4 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν ν2 + ξ 2 ξ + ν2 + ξ 2

dξ

> > > ;

9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > 2 ; 2x ξ − ξ ν2 + ξ 2 ξ + ν2 + ξ 2

(B:12:26)

9 > > =

ðx − ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; 2x ξ − ξ 2 ν2 + ξ 2 ξ + ν2 + ξ 2

(B:12:27)

9 8  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > x > > ð − ξ = < cosh x 1 − e x ν+1 I1 ðxÞ =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi > π ν ! ∞> > > ; : eξ − 1 ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:28)

9 8 > > 2ðx > > μ − 1=2 = < ex ð2 x ξ − ξ 2 Þ ν + 1 Iμ ðxÞ = pffiffiffiffi  lim ν dξ   q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν μ > π Γ μ + 21 ð2xÞ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 Re μ > −

(B:12:25)

(B:12:29)

1 2

9 8 > > ∞ > > ð 2 = < e−x e − ξ =8 x ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ K1=4 ðxÞ = pffiffiffiffiffi lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffi > 2 x ν ! ∞> > > ; : ξ ν2 + ξ 2 ξ + ν2 + ξ 2 0

−x

8 > >
> :

∞ ð

x

− ξ 2 =8 x

(B:12:30) 9 > > =

e qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi > ; ξ ξ 2 − x2 ν2 + ξ 2 ξ + ν2 + ξ 2

(B:12:31)

304

Appendix B Limits Representing Special Functions

8 > >
>

K0 ðxÞ =

:

8 > >
> :

∞ ð

0

9 > > =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 + 2x ξ ν2 + ξ 2 ξ + ν2 + ξ 2 9 > > =

2

e − x =4 ξ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ >; x≠0 > ; ξ ν2 + ξ 2 ξ + ν2 + ξ 2

(B:12:33)

9 8 h  i > > ∞ pffiffiffi −1 ξ > > ð = < sin ν sin ν 2x ffi K0 ðxÞ = dξ lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi > sinh x ν ! ∞> > ; : 0 ξ ðξ 2 + 4x2 Þ ξ + ξ 2 + 4x2 ν2 − ξ 2 >

K0 ðxÞ = e − x lim

8 > >
>

K0 ðxÞ = x e − x

:

νν + 1

∞ ð

0

h

−1



ξ x

i

K0 ðxÞ = e − x lim

ν ! ∞> >

:

νν + 1

∞ ð

0

cosh μ cosh 1+ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:12:35)

(B:12:36)

9 h  i > ξ > = cosh 1 + x dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2 −1

9 8 h  i > > ð cos ν sin − 1 ξ = < ∞ ν K0 ðxÞ = lim ν qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ ; ν ! ∞> > : t 2 + ξ 2 ν2 − ξ 2 ; 0

(B:12:37)

x>0

(B:12:38)

9 8 h  i > > ð ξ sin ν sin − 1 ξ = < ∞ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ K0 ðxÞ = lim ν 3=2 ν ! ∞> > 2 2 : ν2 − ξ 2 ; 0 ðξ + x Þ

(B:12:39)

8 h  i  9 ∞ > > ð sin ν sin − 1 ξ sinh − 1 ξ = < ν x 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi K0 ðxÞ = dξ lim ν > π ν ! ∞> ; : ξ 2 + x 2 ν2 − ξ 2 0

K1 ðxÞ = e − x lim

8 > >
>

:

νν + 1

∞ ð

0

(B:12:34)

9 > > =

9 8 qffiffi > > ξ −1 > > ∞ > > ð sinh = < x ν+1 lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q ν ! ∞> > > > > > ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ; : 8 > >
> =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 + 2 x ν2 + ξ 2 ξ + ν2 + ξ 2

(B:12:41)

305

Appendix B Limits Representing Special Functions

9 8 > > ∞ > > ð = < 1 ξ ν+1 K1 ðxÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi > x ν ! ∞> > > ; : ξ 2 − x2 ν2 + ξ 2 ξ + ν2 + ξ 2 x

K1 ðxÞ =

K1 ðxÞ =

K1 ðxÞ =

8 > >
> : 8 > >
> : 8 > >
> :

∞ ð

0

∞ ð

0

∞ ð

0

9 > > =

ðx + ξÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 + 2x ξ ν2 + ξ 2 ξ + ν2 + ξ 2 − x2 =4 ξ

23=2

(B:12:43)

9 > > =

e qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2 − x2 =4 ξ

(B:12:44) 9 > > =

e qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 ν2 + ξ 2 ξ + ν 2 + ξ 2

9 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 > > 2 ∞ > > ð sin ξ −x = < 1 ν+1 K1 ðxÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q > x ν ! ∞> > > > > ν2 + ξ 2 ξ + ν2 + ξ 2 x ; :

K1 ðxÞ =

(B:12:42)

(B:12:45)

(B:12:46)

9 8 h  i > > > > ∞ ξ > > = < ð cos ν sin − 1 ν x2 lim ν dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 qffiffiffiffiffiffiffiffiffiffiffiffiffi > sinh x ν ! ∞> > > > ; : 0 ξ ξ 2 + 4x2 ξ + ξ 2 + 4 x2 ν2 − ξ 2 >

x>0 (B:12:47) 9 8 h  i  > > ð cos ν sin − 1 ξ = < ∞ 2 Γ μ + 21 x ν qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ pffiffiffi Kμ ðxÞ = lim ν μ + 1=2 ν ! ∞> > π : ν2 − ξ 2 ; ðξ 2 + x2 Þ 0

Re μ > −

1 2

9 8 > > ∞ > > pffiffiffiffi − x ð μ − 1=2 = < 2 ðξ + 2 xξÞ πe ν+1  ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ Kμ ðxÞ = q ffiffiffiffiffiffiffiffiffiffiffiffiffi μ νlim 1 > Γ μ + 2 ð2xÞ ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 Re μ > −

(B:12:48)

1 2

(B:12:49)

306

Appendix B Limits Representing Special Functions

9 8 > > ∞ > > pffiffiffiffi − x ð μ − 1=2 = < 2 2 ðξ − x Þ πe ν+1  Kμ ðxÞ = ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi μ νlim 1 > Γ μ + 2 ð2xÞ ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 x Re μ > −

(B:12:50)

1 2

Kμ ðxÞ =

8 hpffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 μ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 μ i 9 > > ∞ > > ð = < ξ + 2x + ξ − ξ + 2x − ξ e−x ν+1 lim ν dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 2μ + 1 μ xμ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:51)

Kμ ðxÞ =

8 hpffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 μ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 μ i 9 > > ∞ > > ð = < ξ +x + ξ −x − ξ +x − ξ −x e−x ν+1 lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 2μ + 1 μ x μ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 x (B:12:52)

Kμ ðxÞ =

xμ 2μ + 1

Kμ ðxÞ = lim

9 8 > > ∞ > > ð 2 = < e − x =4 ξ ν+1 lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > > > μ+1 ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0 ξ 8 > >
>

Kμ ðxÞ =

:

νν + 1

8 > >
> :

(B:12:53)

9 h  i > > = cosh μ cosh − 1 ξx qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 − x 2 ν2 + ξ 2 ξ + ν 2 + ξ 2 ∞ ð

x

(B:12:54)

9 h  i > −1 ξ > = sinh μ cosh x qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ 2 − x 2 ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:12:55)

9 8 h  i > > ∞ ξ − 1 > > ð = < sinh μ cosh 1+ x e −x ν+1 Kμ ðxÞ = dξ   lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > μ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

Kμ + 1=2 ðxÞ =

pffiffiffi π Γðμ + 21Þ ð2 xÞ

8 > > < lim

μ + 1=2 ν ! ∞>

> :

νν + 1

∞ ð

0

(B:12:56)

2 μ

9 > > =

ðξ 2 − x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:12:57)

307

Appendix B Limits Representing Special Functions

1  I 2 xμ cos π2μ ( 8 μ qffiffiffiffiffiffiffiffiffiffiffiffiffi μ ) 9 h  i qffiffiffiffiffiffiffiffiffiffiffiffiffi > > ξ 2 2 − 1 > > 2 2 > > x +ξ +ξ + x +ξ −ξ > > ν = < ðx cos ν sin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dξ I = lim ν ν!∞ > > > > ðν2 − ξ 2 Þ ðx2 + ξ 2 Þ > > > > ; : 0

Kμ ðxÞ =

jRe μj < 1 (B:12:58) 9 8 h  i ∞  > > ð cos ν sin − 1 ξ μ = < 1 Γ μ + 2 ð2 xÞ ν pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ Kμ ðxÞ = lim ν μ + 1=2 ν ! ∞> > π : ν2 − ξ 2 ; ðξ 2 + x2 Þ 0

x > 0;

Re μ > −

(B:12:59)

1 2

9 8 h  i ∞ 3 > ð sin ν sin − 1 ξ ξðx2 + ξ 2 Þμ − 1=2 > = < 2Γ −μ ν qffiffiffiffiffiffiffiffiffiffiffiffiffi ν dξ Kμ ðxÞ = pffiffiffi 2 μ lim > π ð2xÞ ν ! ∞> ; : ν2 − ξ 2 0

x>0 9 8  2 > > ∞ > > x ð = < Ci 4 ξ 1 ν+1 kerðxÞ = − lim ν dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > 2 ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:60)

Re μ > − 1;

kerμ ðxÞ =

xμ 2μ + 1

(B:12:61)

9 8  > > 2 ∞ πμ μ −1 > > x ð = < ξ cos 2 + 4 ξ ν+1 lim ν dξ   q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:62)

Re μ > − 1

keiμ ðxÞ = −

xμ

8 > > < lim

2μ + 1 ν ! ∞ > > :

νν + 1

∞ ð

0

μ −1



πμ 2

x2 4ξ



9 > > =

ξ sin + qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:12:63)

Re μ > − 1

H − 1 ðxÞ = Y − 1 −

x 4π3=2

8 h   i 9 > > ∞ > > ð ex2 =8 ξ K1 x2 − K0 x2 = < 8ξ 8ξ ν+1 lim ν dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:64)

308

Appendix B Limits Representing Special Functions

8 > >
> :

∞ ð

0

9 pffiffiffiffi  ξ 2 > > = ξ J1=4 4 x qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

9 8  2 pffiffiffiffi > > ∞ ξ > > ð = < ξ J − 1=4 4 x 1 ν+1 H1=4 ðxÞ = Y1=4 ðxÞ + pffiffiffiffi lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi > π x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:65)

(B:12:66)

9 8 > > > > ∞ > > ð = < 2 1 ν+1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H0 ðxÞ = Y0 ðxÞ + dξ lim ν   ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν  2 > π x ν ! ∞> > > > > 0 ; : ν2 + ξ 2 ξ + ν2 + ξ 2 1 + ξx (B:12:67) "

# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ξ + 1+ x

9 8 > > ξ > > > > ln ∞ > > ð x = < 2 ν+1 H0 ðxÞ = Y0 ðxÞ + dξ   lim ν q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ν > π ν ! ∞> > > > > ν2 + ξ 2 ξ + ν2 + ξ 2 0 > > ; :

H0 ðxÞ = Y0 ðxÞ +

2

8 > > < lim

π3=2 ν ! ∞> > :

νν + 1

∞ ð

0

− x2 =8 ξ

 2

K0 8x ξ

(B:12:68)

9 > > =

e  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffi > > 2 2 ; ξ ν + ξ ξ + ν2 + ξ 2

(B:12:69)

2 + Y1 ðxÞ + πx 9 8 > > ∞ > > ð = < 2x 1 ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν > π ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 ξ + x2 + ξ 2 0

(B:12:70)

2 + Y1 ðxÞ + π 9 8 > > ∞ > > ð = < 2 ξ ν+1 dξ   lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > π x ν ! ∞> > > ; : ξ 2 + x2 ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:71)

H1 ðxÞ =

H1 ðxÞ =

9 8 h  i > > ðx ðx2 − ξ 2 Þμ − 1=2 sin ν sin − 1 ξ = < ν 2 ffiffiffiffiffiffiffiffiffiffiffiffi ffi q Hμ ðxÞ = pffiffiffi  lim ν dξ μ > π Γ μ + 21 ð2 xÞ ν ! ∞> ; : ν2 − ξ 2 0

(B:12:72)

Appendix B Limits Representing Special Functions

Hμ ðxÞ = Yμ ðxÞ +

2μ − 1

8 > >
> :

1 μ> − , 2

∞ ð

0

309

9 > > =

2 μ − 1=2

ðx2 + ξ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

(B:12:73)

x>0

H − μ ðxÞ = Y − μ ðxÞ + 9 8   > > μ − 1 x2 =4 ξ > > x ffiffi ∞ > > p ð ξ e erfc = < 2 ξ 2μ cosðπ μÞ ν+1 ; lim ν dξ   ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ν ! ∞> > π xμ > > > > ν2 + ξ 2 ξ + ν2 + ξ 2 0 ; :

Re μ > −

1 2

(B:12:74)

H − μ ðxÞ = Y − μ ðxÞ + 2

μ+1

9 8 ∞  μ > > ð = < 1 Γ μ + 2 x cosðπ μÞ e − ν ξ ν sinh ξ pffiffiffi lim ν h dξ i μ + 1=2 ν ! ∞: > > π ; x2 + ðν sinh ξÞ2

(B:12:75)

0

−

1 3 < Re μ < 2 2

L0 ðxÞ = I0 ðxÞ −

8 > >
> :

∞ ð

0

9  > > = sin − 1 ξx qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ >; > ; ν2 + ξ 2 ξ + ν 2 + ξ 2

9 8 h  i ∞ > > ð sin ν sin − 1 ξ = < ν 2 L0 ðxÞ = I0 ðxÞ − lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ > π ν ! ∞> : ξ 2 + x2 ν2 − ξ 2 ; 0 9 8  2 ∞ > > − x2 =8 ξ x ð = < e I 0 8ξ 1 L0 ðxÞ = I0 ðxÞ − pffiffiffi lim ν pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ > π ν ! ∞> : ξ ξ 2 + x 2 ν2 − ξ 2 ; 0 L0 ðxÞ = K0 ðxÞ + ln x I0 ðxÞ − 8 h  i  qffiffiffiffiffiffiffiffiffiffiffiffiffi  9 −1 ξ > > ∞ > > ln ξ + ξ 2 + x2 = < ð sin ν sin ν 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi dξ lim ν > π ν ! ∞> > > ξ 2 + x 2 ν2 − ξ 2 ; : 0

x>0

(B:12:76)

(B:12:77)

(B:12:78)

(B:12:79)

310

Appendix B Limits Representing Special Functions

9 8 h  i pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ∞ > ð cos ν sin − 1 ξ Y0  2x ξ K0  2 x ξ > = < ν 4 qffiffiffiffiffiffiffiffiffiffiffiffiffi L0 ðxÞ = I0 ðxÞ − lim ν dξ > π ν ! ∞> ; : ν2 − ξ 2 0

(B:12:80) 2 L1 ðxÞ = − + I1 ðxÞ + π 9 8 > > > > ðx = < 2 ξ ν+1  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi > π x ν ! ∞> > > ; : x 2 − ξ 2 ν2 + ξ 2 ξ + ν 2 + ξ 2 0

L1 ðxÞ =

8 > >
> : 0

9  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > − ξ = sinh x 1 − e dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > pffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi2 > ; eξ − 1 ν2 + ξ ξ + ν2 + ξ 2

Lμ ðxÞ = Iμ ðxÞ −

2μ − 1

(B:12:81)

8 > >
> =

1 ðx2 − ξ Þ lim νν + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ pffiffiffiffi  1 μ ν ! ∞ > > π Γ μ+ 2 x > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

1 Re μ > − , 2

(B:12:83)

x>0

Jμ ðxÞ = Jμ ðxÞ + 9 8 qffiffiffiffiffiffiffiffiffiffiffi μ > > 2 > > ∞ > > ð ξ +1−ξ = < sinðπ μÞ ν+1 dξ lim ν   ffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffi λ ν ! ∞> > π > > > > 0 ; : ξ 2 + 1 ν2 + ξ 2 ξ + ν 2 + ξ 2

Jμ ðxÞ = Jμ ðxÞ −

Jμ ðxÞ = Jμ ðxÞ +

8 > > >
πμ > > :

8 > > >
> > :

∞ ð

0

∞ ð

0

(B:12:84)

 qffiffiffiffiffiffiffiffiffiffiffiffiffi− μ  9 > 2 > −μ 2 > ξ + x +ξ −x = qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiλ dξ > > > ; ν2 + ξ 2 ξ + ν 2 + ξ 2



−1 ξ e − μ sinh x

9 > > > =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiλ dξ > 2 > > ; ðξxÞ + 1 ν2 + ξ 2 ξ + ν2 + ξ 2

(B:12:85)

(B:12:86)

311

Appendix B Limits Representing Special Functions

J − μ ðxÞ = J − μ ðxÞ −

"

8 > > > >
> > > :

∞ ð

#μ  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ξ ξ +1 x + x

9 > > > > =

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi   ξ x

0

2

+1

ν2 + ξ 2

(B:12:87)

qffiffiffiffiffiffiffiffiffiffiffiffiffiλ dξ > > > > ξ + ν2 + ξ 2 ;

Eμ ðxÞ = − Yμ ðxÞ + 8   qffiffiffiffiffiffiffiffiffiffiffiμ qffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffi μ  9 2 2 > > ξ ξ2 > > ∞ > > ð 1 + ξx2 − ξx − x + 1 + x2 + cosðπ μÞ 1 + ξx2 = < ν+1 dξ x lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν ffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν ! ∞> > 2 > > > > 1 + ξ 2 ν2 + ξ 2 ξ + ν2 + ξ 2 0 ; : x

(B:12:88)

S − 1, μ ðxÞ =

S − 1 , μ ðxÞ = 8 > > > >
> >
> > :

∞ ð

0

1 S0, μ ðxÞ − μ

1 lim νν + 1 μx ν ! ∞> > > > :

9 > > > =

h  i sinh μsinh − 1 ξx rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > > ; ν2 + ξ 2 ξ + ν2 + ξ 2 1 + ξx

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1− x x

∞ ð

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi  0

1+

ξ x

2

ν2 + ξ 2 ξ +

9 > > > > =

(B:12:90)

qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > > > ν2 + ξ 2 ;

9 8 h  i > > ∞ −1 ξ > > ð = < sinh μ sinh x 1 ν+1 S0, μ ðxÞ =  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ lim ν q ffiffiffiffiffiffiffiffiffiffiffiffiffi > μx ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 9 8 h  i > > > > ∞ ξ − 1 > > ð = < cosh μ sinh x 1 ν+1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S0, μ ðxÞ = lim ν dξ   ffiffiffiffiffiffiffiffiffiffiffiffiffi q q ffiffiffiffiffiffiffiffiffiffiffiffiffi ν  2 > x ν ! ∞> > > > > 0 ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 1 + ξx 8 > > > ∞ > ð < 1 S 0 , μ ðxÞ = lim νν + 1 2μ ν ! ∞> > > 0 > :

(B:12:89)

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1+ x − x

(B:12:91)

(B:12:92)

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1− x x

qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν ν2 + ξ 2 ξ + ν2 + ξ 2

9 > > > > = dξ

> > > > ;

(B:12:93)

312

Appendix B Limits Representing Special Functions

S 1 , μ ðxÞ =

8 > > > >
> > > :

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1+ x + x

∞ ð

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1− x x

qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν ν2 + ξ 2 ξ + ν 2 + ξ 2

0

9 > > > > = dξ

> > > > ; (B:12:94)

9 8 h  i > > ∞ > > ð cosh μ sinh − 1 ξ = < x 1 ν+1 S1, μ ðxÞ = lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ q ffiffiffiffiffiffiffiffiffiffiffiffiffi > x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν2 + ξ 2 0

(B:12:95)

9 8 h  i > > ∞ > >   ð sinh μ sinh − 1 ξ = < x 1 ν+1   S2, μ ðxÞ = x + μ − lim ν dξ qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > μ ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 Sμ − 1, μ ðxÞ =

xμ 2μ + 1

(B:12:96)

( ∞ ð − ν ξ + x2 =4 q ) e lim ν dξ ; q = ν sinh ξ ν!∞ qμ + 1

(B:12:97)

0

Reμ > − 1 8 > > > ∞ > ð < 1 ν+1 s − 1 , μ ðxÞ = lim ν 2μ ν ! ∞> > > 0 > :

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1+ x − x

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  μ ξ ξ +1− x x

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi   ξ x

2

+1

ν2 + ξ 2 ξ +

qffiffiffiffiffiffiffiffiffiffiffiffiffiν ν2 + ξ 2

9 > > > > = dξ

> > > > ;

(B:12:98) 9 8 > > ∞ > > ð 2 = < 1 e − 4 ξ =27 x ν+1 s 0 , 1=3 ðxÞ = lim ν dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > x ν ! ∞> > > ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0

s 2 , 0 ðxÞ = lim

8 > > >
>

> :

νν + 1

∞ ð

0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 > > > − ξ ln ξ x + 1 + ðξ xÞ2 = r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dξ   qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν 2 > > ξ > 2 + ξ2 ξ + ; ν ν2 + ξ 2 + 1 x

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + ð ξ x Þ2 x



9 8  2 > > 2 =8 ξ ∞ > > − x x ð = < e Kμ=2 8 ξ 1 ν+1 s 0 , μ ðxÞ = pffiffiffi lim ν  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ ffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffi > 2 π ν ! ∞> > > ; : ξ ν2 + ξ 2 ξ + ν2 + ξ 2 0 jReμj
>
> : Re μ
> =

1 qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiν dξ > > ; ξ μ + 1=2 ðx2 + ξ 2 Þ ν2 + ξ 2 ξ + ν2 + ξ 2

9 8 h  i μ − 1=2 > > = < ðx cos ν sin − 1 ξν ξ ðξ 2 − x2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi lim ν dξ ν!∞ > > ; : ν2 − ξ 2 0

Re μ > −

313

(B:12:102)

(B:12:103)

1 2

9 8 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ > > ð = < Jμ ðx 1 − e − ξ Þ ν+1 dξ   sμ, 1 − μ ðxÞ = lim ν qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ν > ν ! ∞> > > − ξ Þμ=2 ; : ν2 + ξ 2 ξ + ν 2 + ξ 2 0 ð1 − e

(B:12:104)

Appendix C Notation and Definitions of Special Functions Mathematical Constants π = 3.14159265 . . . γ = 0.577215664 . . . Euler’s constant G = 0.915965594 . . . Catalan’s constant

Exponential Integral and Related Functions ∞ ð

EiðxÞ = −

e−t dt = liðex Þ t

−x ∞ ð

Eið− xÞ = − x ∞ ð

En ðxÞ =

,

x>0

∞ X e−t xn ð− 1Þn dt = γ + ln x + t n!n n=1

e − xt dt, tn

x > 0,

,

n = 0, 1, 2, ...

1

ðx liðxÞ =

1 dt, ln t

x>0

0

liðxÞ = γ + lnðln xÞ +

∞ X ðln xÞn n=1 ∞ X

liðxÞ = γ + lnð− ln xÞ +

n!n

= Eiðln xÞ,

ðln xÞn , n!n n=1

x>1

0 0

0

Γðn + 1Þ = n! ∞ ð

Γðα, xÞ =

tα − 1 e − t dt,

Re α > 0

x

ðx γðα, xÞ =

tα − 1 e − t dt

0

ð1 Bðx, yÞ =

tx − 1 ð1 − tÞy − 1 dt =

0

ψðzÞ =

ΓðxÞΓðyÞ = Bðy, xÞ Γðx + yÞ

d ln ΓðzÞ dz

Error Functions and Fresnel Integrals ðz 2 2 erfðzÞ = pffiffiffi e − t dt π 0

erfcðzÞ = 1 − erfðzÞ ∞ ð

in erfcðzÞ =

in − 1 erfcðtÞdt,

z

2 2 i − 1 erfcðzÞ = pffiffiffi e − z π i0 erfcðzÞ = erfcðzÞ

n = 0, 1, 2...

Appendix C Notation and Definitions of Special Functions



ðz CðzÞ =

cos

 π 2 t dt 2

0



ðz SðzÞ =

sin

 π 2 t dt 2

0

Legendre Functions     1 x + 1 μ=2 1−x − ν, ν + 1; 1 − μ; , j xj < 1 F 2 1 Γð1 − μÞ x − 1 2   π Γðν + μ + 1Þ − μ Pνμ ðxÞ cosðπμÞ − Qμν ðxÞ = Pν ðxÞ 2 sinðπμÞ Γðν − μ + 1Þ

Pνμ ðxÞ =

Complete Elliptic Integrals EðkÞ =

π=2 ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − k2 ðsin θÞ2 dθ =

0 π=2 ð

KðkÞ = 0

  π 1 1 2 F , − ; 1; k 2 1 2 2 2

  dθ π 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2 F1 , ; 1; k2 2 2 2 1 − k2 ðsin θÞ2

Parabolic Cylinder and Whittaker Functions Dμ ðzÞ = 2μ=2 + 1=4 Wμ=2 + 1=4, − 1=4



z2 2



1 Mν, μ ðzÞ = zμ + 1=2 e − z=2 1 F1 ðμ − ν + ; 2μ + 1; zÞ 2 Wν, μ ðzÞ =

Γð− 2μÞ Γð2μÞ Mν, μ ðzÞ + 1 Mν, − μ ðzÞ Γð21 − μ − νÞ Γð2 + μ − νÞ

317

318

Appendix C Notation and Definitions of Special Functions

Jacobi, Gugenbauer, Chebyshev, Legendre, Laguerre and Hermite Orthogonal Polynomials Pnα, β ðxÞ = Cnα ðxÞ =

o dn n α β 2 n ð1 − xÞ ð1 + xÞ ð1 − x Þ 2n n!ð1 − xÞα ð1 + xÞβ dxn ð− 1Þn

o dn n 2 α − 1=2 2 n ð1 − x Þ ð1 − x Þ n 2n n!Γð2αÞΓðα + n + 21Þð1 − x2 Þα − 1=2 dx ð− 1Þn Γðα + 21ÞΓð2α + nÞ

pffiffiffi 1=2 o ð− 1Þn πð1 − x2 Þ dn n n − 1=2 ð1 − x2 Þ 1 n n dx 2 Γðn + 2Þ p ffiffiffi o ð− 1Þn πðn + 1Þ dn n 2 n + 1=2 ð1 − x Þ Un ðxÞ = n 2n + 1 Γðn + 3Þð1 − x2 Þ1=2 dx

Tn ðxÞ =

2

o ð− 1Þ d n 2 n ð1 − x Þ Pn ðxÞ = n 2 n! dxn n

n

x − α ex d n n + α − x fx e g n! dxn o n n 2 d 2 Hn ðxÞ = ð− 1Þn ex e−x n dx dn n 2 o 2 Hen ðxÞ = ð− 1Þn ex =2 n e − x =2 dx Lαn ðxÞ =

Riemann zeta, Lerch, Jonquière, Liouville, Mittag–Leffler and Möbius Functions ζ ðzÞ =

∞ X 1 , z n n=1

ζ ðz, αÞ =

∞ X

1 , ðα + nÞz n=1

Φðz, s, αÞ = Fðz, sÞ =

∞ X

zn , ðα + nÞs n=0

∞ n X z n=1

λðnÞ =

Re z > 1

n

Re z > 1 jzj < 1,

= zΦðz, s, 1ÞFðz, sÞ = s

α ≠ 0, − 1 ∞ n X z n=1

  1 1 ζ ðnÞ, = 1 − 2n ð2m + 1Þn m=0 ∞ X

ns

= zΦðz, s, 1Þ

n = 2, 3, ...

Appendix C Notation and Definitions of Special Functions

Eα ðzÞ =

∞ X

zn Γðαn + 1Þ n=0

Eα, β ðzÞ =

∞ X

zn n!Γðαn + βÞ n=0

m Eα, β ðzÞ =

∞ X

zn , ðn!Þ Γðαn + βÞ n=0 m

m = 1, 2, 3...

the Möbius function μ(n) is defined by μ(n) = 1 if n = 1. μ(n) = (−1)k if n is product of k distinct primes. μ(n) = 0 if n is divisible by a square bigger than unity.

Bernoulli and Euler Polynomials and Numbers ∞ X tezt tn B ðxÞ = , n et − 1 n = 0 n!

Bn = Bn ð0Þ ∞ X 2ezt tn E ðxÞ = n et + 1 n = 0 n!   1 En = 2n En 2

jtj < 2π

319

Index Airy functions – differential and integral relations 12–23 – estimation of zeros as a function of argument 193–211 – integrals 14–16 – series expansions 23–27 Anger functions – differential and integral relations 20–21 – first derivatives with respect to order 86–101 – integral representations 109, 130, 144 – integration with respect to order 86 – recurrence relations 20–21 – second derivatives with respect to order 101–119 – series expansions 25 – third and higher order derivatives with respect to order 119–133 – Weber functions 162–168 Bessel functions – Bessel - Clifford functions 11, 217 – Bessel differential equations 43–77 – Bessel function of the first kind 142–154 – Bessel function of the second kind 93–101 – Bessel functions of equal or nearly equal order and argument 211–215 – Bessel functions of half odd integer order 27–31 – Bessel functions of integer order 86–93 – Bessel functions in terms of special functions 31–33 – Bessel integrals 22–23 – Bickley functions 12 – differential and integral relations 22–23 – estimation of zeros as a function of argument 193–211 – estimation of zeros as a function of order 183–193 – first derivatives with respect to order 90–101 – Hankel function of the first kind 86–101 – Hankel function of the second kind 101–119 – Inequalities 194–196, 200–204, 206, 209, 211, 213, 214 – integral representations 33–43 – integration with respect to order 142–144 – Laplace transforms of Bessel functions 142–154 https://doi.org/10.1515/9783110681642-009

– moments of Bessel functions 138, 140, 152 – Nicholson integrals 101 – Ramanujan integral 134–141 – recurrence relations 12–14, 22–23 – repeated integrals of the Bessel functions 11, 12 – second derivatives with respect to order 119–123 – third and higher order derivatives with respect to order 123–133 – Schlaefli integral representations 106, 107, 110, 115, 123, 130, 138, 146, 180 – series expansions 23–27 Convolution integrals 142, 149, 158, 175, 176, 178, 183 Euler-Rayleigh sums 194, 206–208, 210 Integral Bessel functions – differential and integral formulas 22–23 – first derivatives with respect to order 131, 182–183 – integral representations 104, 106 – integration with respect to order 101 – recurrence relations 22–23 – second derivatives with respect to order 131–133 – series expansions 25–26 – higher order derivatives with respect to the order 131 Kelvin functions – differential and integral relations 21–22 – first derivatives with respect to order 115 – high order derivatives with respect to order 115–117 – integral representations 115, 117 – recurrence relations 21–22 – Scorer functions 9, 10 – series expansions 25 Lommel functions – derivatives with respect to order 86, 92, 128 – differential and integral relations 18–19

322

Index

– first derivatives with respect to order 128 – integral representations 118 – recurrence relations 18–19 – series expansions 24–25, 27 Modified Bessel function of the first kind – differential and integral relations 14–16 – first derivatives with respect to order 90–91, 93–94 – integral representations 108, 115, 157 – integration with respect to order 154–162 – modified Bessel function of the second kind 79–81, 91, 93–94, 154 – recurrence relations 14–16 – second derivatives with respect to order 5, 126 – series expansions 24, 26 – third and higher order derivatives with respect to order 126

Struve functions – differential and integral formulas 16–18 – first derivatives with respect to order 94 – Hankel and Struve transforms 233 – integral representations 111 – integration with respect to order 168 – modified Struve function of the second kind 171–175 – recurrence relations 84 – second derivatives with respect to order 94 – series expansions 81–82 – third and higher order derivatives with respect to order 129 Shifted Dirac function – Lamborn formula 217, 218, 224, 225, 238, 240 – limit representations of special functions and functional series 220–222, 229 Volterra functions 170, 171, 174, 298–299