Polylogarithms and associated functions [1 ed.] 0444005501, 9780444005502

Table of contents :
Title Page
Table of Contents
Foreword
Preface
Preface to the Previous Work
CHAPTER 1. THE DILOGARITHM
1.1. Introduction and Elementary Considerations
1.2. Extension to Large Real Values of z
1.3. Functional Equations Involving a Single Variable
1.4. Numerical Relations
1.5. Functional Relations Involving Two Variables
1.6. Newman's Functional Equation
1.7. Functional Equations Involving Several Variables
1.8. Legendre's Chi-function
1.9. Some Miscellaneous Results
1.10. A Survey of Definitions and Notations
1.11. Relations to Other Mathematical Functions
1.12. Occurrence in Physical Problems
CHAPTER 2. THE INVERSE TANGENT INTEGRAL
2.1. Elementary Considerations and Definition
2.2. The Inversion Relation
2.3. The Duplication Formula
2.4. Some Numerical Relations
2.5. The Triplication Formula
2.6. The Multiplication Formula for Odd Multiples
2.7. The Quadruplication Formula
2.8. Functional Equations Involving Several Variables
CHAPTER 3. THE GENERALIZED INVERSE TANGENT INTEGRAL
3.1. Introduction and Elementary Properties
3.2. Differentiation with Respect to the Parameter
3.3. Formulas Arising from a Change of Variable
3.4. Formulas Arising from Inverse Tangent Integrals of Bilinear Argument
3.5. Formulas Arising from Inverse Tangent Integrals of Biquadratic Argument
3.6. Factorization Theorems
3.7. Multiplication Formulas
3.8. Derived Relations
3.9. Special Values of the Parameter
3.10. An Addition Equation Involving Argument and Parameter
CHAPTER 4. CLAUSEN’S INTEGRAL
4.1. Definition and Elementary Properties
4.2. Periodic Properties
4.3. The Factorization Theorem
4.4. Series Expansions
4.5. Integral Relations
4.6. Functional Equations
4.7. Geometrical Connections
CHAPTER 5. THE DILOGARITHM OF COMPLEX ARGUMENT
5.1. Resolution into Real and Imaginary Parts
5.2. The Factorization Theorem
5.3. Special Values of the Argument
5.4. Functional Equations Involving a Single Variable
5.5. Reduction of Li[2](x,theta) for Special Values of theta
5.6. Newman’s FunctionaI Equation Involving Two Variables
5.7. Derived Functional Equations
5.8. Consequences of the Duplication Formula
5.9. An Addition Formula for the Angular Parameter
CHAPTER 6. THE TRILOGARITHM
6.1. Introduction and Elementary Considerations
6.2. Functional Equations of a Single Variable
6.3. Numerical Relations
6.4. A Consideration of Some Complex Forms
6.5. A Generalization of Clausen’s Integral
6.6. A Further Consideration of Complex Forms
6.7. Functional Equations of Two Variables
6.8. A Functional Equation of Newman’s Type
6.9. Functional Equations Involving Several Variables
CHAPTER 7. THE HIGHER-ORDER FUNCTIONS
7.1. Introduction and Definitions
7.2. The Inversion Equation and Its Consequences
7.3. The Factorization Theorems
7.4. Associated Integrals
7.5. The Associated Clausen Functions
7.6. Integral Relations for the Fourth-Order Polylogarithm
7.7. Functional Equations for the Fourth-Order Polylogarithm
7.8. Functional Equations for the Fifth-Order Polylogarithm
7.9. The Log-Sine Integrals
7.10. Results from a Contour Integration
7.11. Golden-Cut and Related Integrals
7.12. Polylogarithms of Nonintegral Order
7.13. Higher-Order Polylogarithms
CHAPTER 8. INTEGRATION OF FUNCTIONS AND SUMMATION OF SERIES
8.1. Reduction of a Class of Algebraic and Logarithmic Expressions
8.2. Reduction of Trigonometric Forms
8.3. Summation of Series
8.4. Integrals from the Higher-Order Functions
8.5. Definite Trigonometric Integrals
APPENDIX. REFERENCE DATA AND TABLES
A.1. Glossary of Notation
A.2. List of Selected Formulas
A.3. Reference List of Integrals
A.4. Tabulated Values
Bibliography
Suggestions for Further Study
Index

Citation preview

Leonard Lewin· Polylogarithms and Associated Functions

CONTENTS

vii

CHAPTER 7. THE HIGHER-ORDER FUNCTIONS 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13.

Introduction and Definitions The Inversion Equation and Its Consequences The Factorization Theorems Associated Integrals The Associated Clausen Functions Integral Relations for the Fourth-Order Polylogarithm Functional Equations for the Fourth-Order Poly logarithm Functional Equations for the Fifth-Order Poly logarithm The Log-Sine Integrals Results from a Contour Integration Golden-Cut and Related Integrals Poly logarithms of Nonintegral Order Higher-Order Polylogarithms

CHAPTER 8. INTEGRATION OF FUNCTIONS AND SUMMATION OF SERIES 8.1. Reduction of a Class of Algebraic and Logarithmic Expressions 8.2. Reduction of Trigonometric Forms 8.3. Summation of Series 8.4. Integrals from the Higher-Order Functions 8.5. Definite Trigonometric Integrals APPENDIX. REFERENCE DATA AND TABLES A.l. A.2. A.3. A.4.

Glossary of Notation List of Selected Formulas Reference List of Integrals Tabulated Values

189 189 192 197 199 200 202 206 212 216 228 232 236 238

242 242 253 260 269 274 281 281 283 303 312

Bibliography

349

Suggestions for Further Study

355

Index

357

POLYLOGARITHMS AND

ASSOCIATED FUNCTIONS Leonard Lewin Department of Electrical Engineering University of Colorado, Boulder, Colorado

~ ~ NORTH HOLLAND New York • Oxford

Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, New York 10017 Sole distributors outside the USA and Canada: Elsevier Science Publishers B. V . P.O. Box 211, 1000 AE Amsterdam, The Netherlands

©

1981 by Elsevier North Holland, Inc.

Library of Congress Cataloging in Publication Data Lewin, Leonard, 1919Polylogarithms and associated functions. Bibliography: p. Includes index. 1. Logarithmic functions. I. Title. QA342.L47 515'.25 80-20945 ISBN 0-444-00550-1 Desk Editor Danielle Ponsolle Design Edmee Froment Design Editor Glen Burris Art rendered by Vantage Art, Inc. Production Manager Joanne Jay Compositor Science Typographers, Inc. Printer Haddon Craftsmen

Manufactured in the United States of America

CONTENTS

Foreword Preface Preface to the Previous Work

IX

xi xiii

CHAPTER 1. THE DILOGARITHM 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1. 7. 1.8. 1.9. 1.10. 1.11. 1.12.

Introduction and Elementary Considerations Extension to Large Real Values of z Functional Equations Involving a Single Variable Numerical Relations Functional Relations Involving Two Variables Newman's Functional Equation Functional Equations Involving Several Variables Legendre's Chi-function Some Miscellaneous Results A Survey of Definitions and Notations Relations to Other Mathematical Functions Occurrence in Physical Problems

CHAPTER 2. THE INVERSE TANGENT INTEGRAL 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

Elementary Considerations ~nd Definition The Inversion Relation The Duplication Formula Some Numerical Relations The Triplication Formula The Multiplication Formula for Odd Multiples The Quadruplication Formula Functional Equations Involving Several Variables

1 2 4 6 7 11 16 18 21 27 30 31 38 38 39 40 44 45 47 51 59

CONTENTS

vi

CHAPTER 3. THE GENERALIZED INVERSE TANGENT INTEGRAL 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10.

Introduction and Elementary Properties Differentiation with Respect to the Parameter Formulas Arising from a Change of Variable Formulas Arising from Inverse Tangent Integrals of Bilinear Argument Formulas Arising from Inverse Tangent Integrals of Biquadratic Argument Factorization Theorems Multiplication Formulas Derived Relations Special Values of the Parameter An Addition Equation Involving Argument and Parameter

68 68 70 74 78 79 89 92 93 94 96

CHAPTER 4. CLAUSEN'S INTEGRAL 4.1. Definition and Elementary Properties 4.2. Periodic Properties 4.3. The Factorization Theorem 4.4. Series Expansions 4.5. Integral Relations 4.6. Functional Equations 4.7. Geometrical Connections

101 101 102 104 105 106 108 115

CHAPTER 5. THE DILOGARITHM OF COMPLEX ARGUMENT

120 120 123 124 126

5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

Resolution into Real and Imaginary Parts The Factorization Theorem Special Values of the Argument Functional Equations Involving a Single Variable Reduction of Li 2(x,8) for Special Values of-8 Newman's Functional Equation Involving Two Variables 5.7. Derived Functional Equations 5.8. Consequences of the Duplication Formula 5.9. An Addition Formula for the Angular Parameter

CHAPTER 6. THE TRILOGARITHM 6.1. Introduction and Elementary Considerations 6.2. Functional Equations of a Single Variable 6.3. Numerical Relations 6.4. A Consideration of Some Complex Forms 6.5. A Generalization of Clausen's Integral 6.6. A Further Consideration of Complex Forms 6.7. Functional Equations of Two Variables 6.8. A Functional Equation of Newman's Type 6.9. Functional Equations Involving Several Variables

131 132 134 137 139 153 153 154 155 158 162 166

172 185 187

CONTENTS

vii

CHAPTER 7. THE HIGHER-ORDER FUNCTIONS 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13.

Introduction and Definitions The Inversion Equation and Its Consequences The Factorization Theorems Associated Integrals The Associated Clausen Functions Integral Relations for the Fourth-Order Polylogarithm Functional Equations for the Fourth-Order Poly logarithm Functional Equations for the Fifth-Order Poly logarithm The Log-Sine Integrals Results from a Contour Integration Golden-Cut and Related Integrals Poly logarithms of Nonintegral Order Higher-Order Polylogarithms

CHAPTER 8. INTEGRATION OF FUNCTIONS AND SUMMATION OF SERIES 8.1. Reduction of a Class of Algebraic and Logarithmic Expressions 8.2. Reduction of Trigonometric Forms 8.3. Summation of Series 8.4. Integrals from the Higher-Order Functions 8.5. Definite Trigonometric Integrals APPENDIX. REFERENCE DATA AND TABLES A.l. A.2. A.3. A.4.

Glossary of Notation List of Selected Formulas Reference List of Integrals Tabulated Values

189 189 192 197 199 200 202 206 212 216 228 232 236 238

242 242 253 260 269 274 281 281 283 303 312

Bibliography

349

Suggestions for Further Study

355

Index

357

CONTENTS

vii

CHAPTER 7. THE HIGHER-ORDER FUNCTIONS 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13.

Introduction and Definitions The Inversion Equation and Its Consequences The Factorization Theorems Associated Integrals The Associated Clausen Functions Integral Relations for the Fourth-Order Polylogarithm Functional Equations for the Fourth-Order Poly logarithm Functional Equations for the Fifth-Order Poly logarithm The Log-Sine Integrals Results from a Contour Integration Golden-Cut and Related Integrals Poly logarithms of Nonintegral Order Higher-Order Polylogarithms

CHAPTER 8. INTEGRATION OF FUNCTIONS AND SUMMATION OF SERIES 8.1. Reduction of a Class of Algebraic and Logarithmic Expressions 8.2. Reduction of Trigonometric Forms 8.3. Summation of Series 8.4. Integrals from the Higher-Order Functions 8.5. Definite Trigonometric Integrals APPENDIX. REFERENCE DATA AND TABLES A.l. A.2. A.3. A.4.

Glossary of Notation List of Selected Formulas Reference List of Integrals Tabulated Values

189 189 192 197 199 200 202 206 212 216 228 232 236 238

242 242 253 260 269 274 281 281 283 303 312

Bibliography

349

Suggestions for Further Study

355

Index

357

PREFACE TO THE PREVIOUS WORK

I attempt to give here some account of what this book is about, and how it came to be written. This has turned out to be a more difficult task than I had imagined since it had its origins over twenty years ago when I was still a pupil at school, and memories of that time have dimmed somewhat. But standing out is the recollection of glancing through the pages of Edward’s Calculus-a fascinating book if ever there was-when my eye caught a paragraph at the foot of the page recounting some curious formulas of Landen’s: the subject cropped up again in the examples. So bizarre are these results that it seems warranted to elevate them here for special mention. Typical is the relation (%A)

N5-1)

/log (1-x) dxlx = )?!(gol

-7T2/10.

0

I wonder how many there must be, professional mathematicians and amateurs, whose imaginations have been fired by this result. It certainly captured mine: it seemed to be pointing to something of great interest, something deeper and more radical than a mere trivial numerical result, but I could not quite grasp what. Definite integrals I had at that time already met; I could understand how, even though the indefinite form might not be resolvable, the integral between, say, zero and infinity, or zero and 27~, could be evaluated. These limits correspond to an entire range, or a period, or some other special feature. But what about ?h(d%l)? Why this peculiar value? What was the significance of this form, and for what other special values was the integral similarly expressible? The whole thing puzzled me greatly and I resolved to try and find out. Fortunately the methods to be used were not at all difficult. I do not think, surveying the subject today, that I ever really solved the problems I set out to solve. Perhaps they were badly formulated, or were not real problems at all. But I soon began discovering all sorts of results. It is an easy subject to explore, rather surprisingly, because

POLYLOGARITHMS AND

ASSOCIATED FUNCTIONS

CHAPTER

1

THE DILOGARITHM

1.1. INTRODUCI’ION AND

ELEMENTARY CONSIDERATIONS

1.1.1. The functions investigated in this chapter arise from a consideration of the series

2 z3 -+-+-+... l Z } S 1. 1’ 2’ 3’ Discussed by Euler’ in 1768, this function was later called the dilogarithm by Hill’ from analogy to an integral formulation, derived as follows: Since

we find, by dividing by z and integrating,

a

;lz*

z3 d z = i z dz -+-+-+... =1-2’ 1’ 2’ 3’ the double integration on the right suggesting the name. Although the series is only convergent for Iz I 1, the integral is not restricted to these limits, and we define for all values of z or -i’log(l-z)-log(z)&. d (1.4) zlog(1-2) dz ~i,(z= ) 2 dz

1

Li,(t) is the dilogarithm function. 1.1.2. The question of notation is an awkward one. Practically every writer who has investigated this function has developed his own, and no two of the early writers agree. Such notations as are used do not lend themselves readily to generalization to the higher-order €unctions discussed later, and

3

FIGURE 1. Branch cut for the dilogarithm

\

\

\i \ \ -

r log x

\

FIGURE 2. The dilogarithm for real variable x -, real part of Li,(x); ---imaginary part Li,(x);

--- asymptotic form for large x.

1.5. FUNCTIONAL RELATIONS INVOLVING TWO VARIABLES

Also, by taking z=(3- v5)/2 in (1.11) we get

(1.19) Solving (1.18) and (1.19) gives

and (1.20) where it may be noted that the argument (v5 - 1)/2 =2 sin( v / 10). 1.43. Taking x=(3- v5)/2 and ( v 5 - 1)/2 in (1.12) gives the two further results

and (1.21) These would appear to be the only real values for which the Li, function can be expressed in terms of more elementary ones. However, there are some interesting relations which can be derived from the dilogarithm of complex argument, and these are given in Section 5.9.5. They involve the dilogarithm of arguments based on trigonometrical functions of angles v / M . The case M = 3 reduces to a trivial identity, M = 4 gives (1.16), and M = 5 gives, after some rearrangement, (1.20). The case M = 6 gives a relation connecting dilogarithms of arguments and f, more easily deduced from (1.12) and (1.15), while the particular case M=7 was first given by Watson in 1937; it can also be found from a formula of a rather different character due to Richmond and Szekeres.* The details are discussed in Chapter 5. 1.5. FUNCTIONAL RELATIONS INVOLVING TWO VARIABLES 1.5.1. Equation (1.12) is the particular case K = - 1 of a more general formula arising from a consideration of the argument Kx/(l - x ) . By u

+G. Szekeres, private communication.

C M E R 1. THE DILOGARITHM

10

1

In this equation L (1 + x ) means log( 1+x). A very similar expression was also found in the same way, starting with

In order to see the relation of (1.28) to the previous formulas, write -X for mx and - Y for nx. Reverting to the Li, notation, Equation (1.28) becomes Li,(X+ Y-XY)=Li,(X)+Li,(Y)-Li XY- Y XY-x

-log( -log(

Y-XY x+ y-xy)log( X 7) x+ ;-xy)log( 7). x-XY (1.29)

In this equation put Y= 1- l/Z: ~ i , 1( -

y)

= L~,(x)

+~ i , ( l - I / Z )

+various logarithmic terms. On comparison with (1.22) it can be seen that the use of (1.7) to invert the argument of certain terms, together with the use of (1.1 I), will bring the two forms together. Thus the same functional equation can be deduced by starting either with a quadratic or a bilinear form. These five-term results stand in contrast to Newman’s six-term formula of Section 1.6. 1.5.4. A large number of single-variable relations may be obtained by taking y as some suitable function of x in the preceding formulas. The following, which are representative, are taken from Nielsen’s monograph.l3 From (1.27) with y = i x

2-2x z) +~ i , z ( )

1-x ~ i , ( x )- ~ i , ( f x )- ~ i , ( =T 2 /

-)”:;(-

12+ ilog2(2) - log x log

(1.30)

This result is due to Schaeffer. From (1.25), with 8 =x, = 2 - x, we appear to get Kummer’s formula

+ ~ i , ( 2 x-x2) =2 ~ i , ( x )+ 2 ~ i , ( 2- x ) - 31.’.

1.6. NEWMAN’S FUNCTIONAL EQUATION

11

This Substitution, however, is faulty, sinee it fails to take accOunt of the form (1.13) when the argument is greater than unity. It is impossible to take 8 = x , + = ( 2 - x ) and still have 8 and +O.

(3.65)

This is identical in form to (3.13). 3 5 5 . A more general formula, which contains the previous results as Special cases, Comes from the equation

tan-'(x)

+tan- '(ax+b) =tan-'

( a + l)x+b 1-bx-ax2

Then

1 [

= tan-'(x)+ tan-'(ax+b)]

1 x+b/(a+l)

1 X-c

1 x-d

9

(3.67)

where c and d are the roots of the equation ax2+bx - 1=0,

c , d = ( -b?

)/2a.

(3.68)

3.9. SPECIAL VALUES OF THE PARAMETER

95

he was missing something is apparent from the quotation from his writings given in Section 3.1.3. In Chapter 5 the Same Situation arises for ReLi,(reie), a function analogous in many ways to Ti,(x, a). There we also deduce values of 8 for which the reduction to simpler functions is possible. Rather significantly the Same values 8= (2n + i ) ~ / 2 " appear at an early Stage of the analysis. But by means of a functional equation involving, among other things, a simple addition to the angular parameter it is possible by an iteration process to achieve a reduction in all cases when 8 / is~ a rational fraction. One gets a very strong feeling that the Same Situation exists for Ti,(x, tan 8) and that somewhere there is an analogous functional equation permitting a similar reduction here. Of the many formulas produced in this chapter none seem to quite foot the bill, though several come rather near to doing so, particularly those of Section 3.3. However, we must admit that to date even the simple value a = tan(7r/6)= l / V 3 has had to be passed over unresolved, and the only successes are the ones foreshadowed in Spence's analysis.

+

3.9.2. We shall indicate here how the caSes with a = tan[(2n 1~ ) / 2 " ] can be reduced. The Set involving tan(n/8) has in fact already been dealt with in an ad hoc manner in Chapter 2 in connection with functional equations for Ti,(x). The More general relation can be obtained from Spence's analysis, but a quicker route is via (3.80). If we there put x=tan8 and a = tan we obtain

+

:

:Ti ,( tan28, tan 2 +) - Ti ,( tan 8, tan +) = V(8 , +),

(3.101)

where V( 8, +) =Ti,(tan28) - 2Ti2(tan8) -2Ti,[ tan(

@++)I

+2 Ti ,(tan +) + 2+1og[ tan( 8++ ) / a n +] .

(3.102)

In (3.101) put 28,228,...,2"-'8 for 8 and a similar set of values for +. Divide successive equations by 4 and add all the results. On the left only the first and the last term remain, all the others canceling. On the right we get a sum of tems 2 -%"+ ')V(2"8,2"+). The resulting equation is 1 Ti,(tane,tan+) = -Ti2[ 2,"

n-1

tan(l"@),tan(2"+)] -

4-"V(2"8,2"'+). 0

(3.103)

+

All tems on the right are known except the first, and if is taken as ~ / 2 " + 'it vanishes since Ti,(x, oo)= 0. Larger multiples of 77/2"+' cannot be used here since some of the V-expressions exhibit singuiarities. How-

3.10. EQUATION INVOLVING ARGUMENT AND PARAMETER

99

When k is either c or e, E takes a very simple form. From the second of the relations of (3.105) we get c - a / ß = -(e+8/ß), and on dividing through by the right-hand side and multiplying by l/c we obtain (3.1 11) Since -ce=y/ß, the left-hand side of (3.111) is just Hc). Similarly, e(e)= l/e. But when k is a or b, a simple form for E does not seem possible. Perhaps the most compact is ( a , b)( 1-ab) -A(ce- ab) + ab( c +e) (3.1 12) E(a, b ) = -ce(l -ab)-(a, b)[ A(ce-ab)-(c+e)] ’ where A is given by (3.107). Using these results to complete the integration of (3.109) the following equation is found:

=Ti2(Y, a ) + Ti2(Y , b) - Ti,( Y , c ) - Ti2(Y , e )

+ T i , [ z , ~ ( a ) ] + T i ~ [ z , ~ ( b-Ti,(z,l/c)-Ti,(z,l/e)+C. )] (3.1 13) In this equation A is given by (3.103, z is given in terms of y by (3.108), and e(a, b) is given by (3.1 12). There are five independent variables, a, b, c, e, and Y , while C must be detennined by using some appropriate value of Y . Unfortunately z does not vanish with y, so that y=O is not a suitable choice. However, if we sacrifice one of the constants by imposing the additional restriction ce[(a+b)(c+e)+(l-ab)(l -ce)] -ab(l+c2)(1 +e2)=0,

(3.114)

then a = O and z vanishes withy. In this case, C=O and (3.1 13) takes on a somewhat simpler form. With a =0, (3.105) gives 8 / ß = - ( c +e). Hence e simplifies considerably, in this case, to give e(a, b)=l/[c+e-ce/(a,b)], In the Same way (3.108) for z becomes z = -y/[ce+(e+c)y].

(3.1 15) (3.1 16)

Lastly, the restriction (3.114) itself can be simplified. From the fourth relation of (3.105), with a = O and 8 / ß = -(c+e) we get A( Ce - ab) =ab( c+ e). (3.1 17)

3.103. How does the resulting equation compare with the target? Because of (3.1 16) the two variables y and z are not unrelated: if y and z are

CHAPTER

4

CLAUSEN'S INTEGRAL

4.1. DEFINITlON AND ELEMENTARY PROPERTIES 4.1.1. The function discussed in this chapter arises from a consideration of

the diiogarithm of imaginary exponential argument. More specifically we cunsider the form Li2(eie), and from the series definition we have 00

Li2(eie)= 1

00

cos n e sin ne -+iC -. n2 1 n2

From the definition as an integral we have also Li2(eie)- ~ i , ( l )= - f i ' Take a new variable Then

log( ;-2) dz.

+ given by t =ei*, so that the iimits for + are 8 and 0.

Li 2( eie ) - Li 2( 1) = -iJ'log(

= -i/'log[

i -ei* ) ti+

2sin(&b)] d++ f(7r-6)'-

fw2,

(4.2)

0

provided 0$6S 2n. On equating the real parts of (4.2), using (4.1) for the left-hand side, we get a well-known Fourier series

4.2. PERIODIC PROPERTIES

103

4.2.2. An alternative definition to (4.5) which permits a consideration of

negative values is

With this definition the sine series and the integral are identical. Accordingly, we have, from the periodic properties of the series, C1,(2nn 2 0) =C1,(

8 ) = 2 C12(0).

Similarly, or from (4.10) with n = 1 and n+0 for 8, ~ i , ( ~ + e )- = ~i,(~-e).

(4.10) (4.1 1)

4.23. Some particular values of C1, will now be obtained. Obviously, from the series, we have (4.12) C1 (n n ) =0.

,

The particular case n = 1 gives jZlog(2 sin; 8) d8-0, which is readily reduced to a more familiar equation iw’2iog(sine ) de = - inlog 2

(4.13)

by putting 28 for 8 and decomposing the logarithm. From the series, with 8= n, we get C12(+)=

1 1 1 7 - 32 + 52 - - - = G , *

FIGURE 4. Plot of Cl,(B) and Gl,(B).

2.0)

-0.4

-0.8 -1.2 -1.6

-2.0

(4.14)

4.5. INTEGRAL RELATIONS

107

Using (4.31) to express the inverse tangent integrals in terms of Clausen's function we get

+

12G=4C12(28)+4Cl,( n - 28) 2C1,( i n -28) +2C12(i n +28)

+ C1,(48) + C12(n-48)

(4.34)

where tan28= $. There does not appear to be any obvious way of deducing this from the factorization theorem for Cl,. However, by application of the duplication formula C1,(2+) =2 C1,( +) -2 C1,( n -+), it can be put in the simpler form 12G-8Cl2(28) +2Cl,(4n+ 28) + 2C12(in-28) - $1,(88), (4.35) where tan28=

:.

4.5.5. From the definition of Ti,(x, a) we get

+

log(x a) dx . 1+x2 Put a = tan+ (+ a constant) and x = tan8. Then sin(8++) x+a= wsews+ and Ti ,( tan 8, tan +) Ti ,( x ,a) = log( x +a) tan- * (x ) - J

x

(4.36)

0

=8log[ sin( cos8 8 ++) ]+~c12(28+2+)+;c12(n-2t9)-;c~2(2+). (4.37) Equation (4.31) is the special case +=O. If it could be solved for C1, in terms of Ti, it would be possible to relate the inverse tangent integral to the generalized integral. But as demonstrated in Chapter 3, it is not likely that this can be done except for special values. It is therefore apparent that (4.31) cannot be reversed to express C1, in terms of Ti,, at least not in closed terms. (An infinite series is readily obtained by iteration but is useless for the present purpose.) 4.5.6. A particular case of (4.37) is 8=+:

+

,

,

Ti 2( tan +,tan +) = log(2 sin +) + 4 Cl (4+ ) + Cl ( T - 2+) - 4 C12(2+) .

CHAPTER 4. CLAUSEN'S iNTEGRAL

110

Rogers' has deduced a very general five-variable formula (with a Single relation between the variables) from an equation involving Li, of complex argumeat, and this we shall now derive.* 4.6.5. As explained in Chapter 5, the imaginary part of the dilogarithm is

expressible in terns of C12, the actual relation h i n g Im Li 2( reie)=o log r + $1 &)

+ $1

2(28) - :Ci ,(SO

+2e),

(4.46)

where tan o =rsin 8 / ( 1 - rcos 8). I€ this fonnula be used indiscriminately On any of the two-variable relations of Chapter f, an equation of sorts for C1, will be prodwed, but because of the form taken by the variables, and in particular by the angle o above, the arguments will be far from simple. However, a particular structure, considered in great detail by Rogers, has a very remarkable cyclic property which permits, as it happens, an extremely symmetrical arrangement of the variables. We Start with Hill's form, Equation (1.24), of the two-variable relation for Li2, and transform by means of (1.1 1) the arguments of the first, fourth, and fifth terms to give their complements. The resulting equation is

(-)=L7 1-xy

L i , ( x ) + ~ i , ( y ) + ~ i( - 1) -+x~ i ~ ( l - x y ) + ~ i 1-xy where L=

+'

- log xlog( 1-x) - log y log( 1-y

(

) + log ;:)log(

(4.47)

2).

If the five arguments O n the left of (4.47) be denoted in Order by z,, n = 1,2.. .5, then the cyclic property 'PI = -'n+2',+3 (4.48) is readily verified (it being understood that t5+,,, is to be interpreted as z,,,). From here O n x and y will be taken to be complex variables. The representation of a complex variable may be in the polar form re", or alternatively it can be given by the bipolar (8, +) or bivector (r, s) Coordinates. The relation between these equivalent fonns is shown in Figure 5. Obviously we have r/sin =s/sin8 = l/sin( 8+ +) . (4.49)

+

We consider the five complex variables z, of (4.47) and represent each by s,, and +., We seek the relations which must the Set of quantities r,, exist between all these quantities.

*Rogen studied the function T(8)= J,8(8/tanO)d@, which can bc simply related to C1,(28) through an integration by parts.

116

CHAFTER 4. CJAUSWS MTEGRAL

When

+

sin’ a sin’ y = sin’ ß,

(4.75)

D = sin a sin Y , so that X = 0 and

S(a,ß,y)=-a2+ß2-y2.

(4.76)

When cos a cos y =cos ß,

(4.77)

D =0, so that X = 1, whence

S(a,ß,y)=O. By taking the differential of (4.70), we get

(4.78)

ao

(-x)“(sin 2na da - sin 2 n ß dfl+ sin 2nydy)

-2 l

n

- 2( ada -ß dß+ y d y ). (4.79) On summing the series the coefficient of dX is found to be Zero, and the differential is reduced to COS a sin y

- idS( a, ß, y ) = cos-

(cos2 a - cos’ 8)”’

1

da

- COS sin a cos y

+cos(COS’

y - cos’ß)1/2

]

dY.

(4.80)

With ß = n / 2 this reduces to dS( a, i n , Y ) = - ( n - 2 y ) d a - ( n - 2 a ) d y ,

(4.8 1)

whence

S( a, in, y ) =2( i n -a)( ir - y ).

(4.82)

With #I f= n in (4.70) we therefore obtain an identity from (4.82) which, with a Change of variables, can be written n

cosnxcosny-cos’inn

=1(1 2 27I-x

)’

?

n’

s

0 6x n , cos2 x