Some generalizations of the Bessel function

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,

This thesis having been approved by the

,

special Faculty Committee is accepted by the Graduate School of the University of Wyoming

,

in partial fulfillm ent of the requirements for the degree of

Masti3r__of__A.rts__ ________ Dean of the Graduate School.

Tint#

August1$,1950

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SOME GENERALIZATIONS OP THE BESSEL FUNCTION by Lawrence J. Prince

A Thesis Submitted to the Department of Mathematics and the Graduate School of the University of Wyoming in Partial Fulfillment of Requirements for the Degree of Master of Arts

University of Wyoming Laramie, Wyoming August, 1950

LIBRARY OF THE

UNIVERSITY OF WYOMING LARAMIE

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UMI N um ber: E P 24226

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ACKNOWLEDGEMENT The writer wishes to express his gratitude to Doctor Nathan Schwid for his invaluable guidance and assistance in the preparation of this thesis.

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TABLE OP CONTENTS Chapter

Pag©

I. . ..........................................

1

Representation of Series Using the Gamma Function •

3

Integral Representation of Jn,m(x)..........

4

Recursion Formulas........................ .

8

The Functions of Integral Orders

............... 11 12

.

Zeros of Jn#m(x).......................

Orthogonality....... • • • • . ............... 18 Evaluation of the Norm. . . . . . . . .

.....

22

Closed Forms of Jn,m(x) ....................... 26 Fourier-Bessel Expansion of Functions . . . . . . .

29

Solution Corresponding to a Bessel Function of Second Kind.

................................32

II..................

41

Representation of Series Using the Gamma Function • 43 Integral Representation of Series

44

Recursion Formulas.........

46

Zeros of Kn,m(x). . . . . . . . . . . . .

.......

48

Orthogonality

50

Evaluation of the Norm. . . . . . . . . . . . . . .

54

SELECTED REFERENCES .

................................57

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INTRODUCTION The purpose of this study is to present two generaliza­ tions of the Bessel function.

A Bessel function is usually

defined as a particular solution of the linear differential equation of the second order, (0.1)

X*

+ (X*-/»/yU = 0 ,

+ X d. x

dx

known as Bessel»s equation.

"

In this equation, as we shall

consider it, n is any real number. When n is neither a positive nor a negative integer, nor zero, the general solution of Bessel's equation has been found to be

^ A J » + 0£/«(*) 1 a

where A and B are arbitrary constants and Jn(x) is the partic­ ular solution found by taking ) mA - O

*

(A

or, if

This is a recursion formula for A^, giving each coefficient in terms of one appearing earlier in the series. Let us make the choice s = n so that the recursion form­ ula becomes

Since Ai - 0, it follows that A3 = 0 , hence As=0, etc., that is,

(1 .6 )

(A *

I,

• •

J

provided n is such that k+ 2n^0 in formula (1.5). Replacing k by 2k in formula (1.5), we can write

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provided n is not a negative integer* Using formula (1*7), we can write the first few coef­ ficients of our series as

and our series can then be written as .xvCi

_________ t L 0 ± a n )l

XXCl*-rt)

+

)(x+/")C3 +" ’)

where n is not a negative integer and m is any real number* The function represented by this series corresponds to the Bessel function of the first kind of order n.

We shall,

hereafter, designate this function as Jn>m(x), when a specific value is assigned to Aq as is done below* The series in the brackets of (1*8) is absolutely con­ vergent for all values of x according to the ratio test. Since it is a power series, the termwise differentiation em­ ployed to arrive at (1*2) was valid and Jn,m(x) ^s» therefore, a solution of the differential equation (1.1)* Representation of Series Using the Gamma Function If we assign to Aq in (1.8), the value

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4 where

T (n + 1) is

the well-known Gamma function, we can repre­

sent Jn,m(x) as

T (%) =

XU

\

m./m

X1'*(!+/*»)--/ (/yn-/)(/H+j)'--(/» +# )

Jk-o

or, since

T(/n*Ji + i) = (/n+i)(fflti)(/n+s) -•(m +A) T ( / » + j) ,

d.9)

✓i .4 / \ *A T.„(x) = M M JuUwxti— kL-^ . k ah

I— -k~ 6

J ! T ( » ' - » * ’)

Integral Representation of Jn-'4 }A > - s*)

or

(1.12)

T ( * * t U -M * *J s TC/n +b + t) J

b h £ M0 A Q 9

(/»>-% * A ? o )

^o

since /* ^ X

A* ?

_ tI

& fa * 0 d $ = \dJM0 C < H l^o d .0 + Ixu#t*/f'&&KL O d o 9

and by making the change of variable 0 s 7T-& in the last in­ tegral of the above equation it becomes

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aimJ/r>0'c # x *0 ,do

-

r*

Ia m

2/”(tt~ 6)

J A*MX/nQ

jr-6 )(-d & )

d O

JZ

where k is a positive integer or zero.

Hence

**0 CM?*# d 6 = ^ A tM X/>’0 O/XL^O d o .

(Jl 2 o)

0

We will now find it convenient to write

(1 13)

I

(x ) - /*f

(->)*X *

-*.-o Multiplying and dividing the fraction,

by %■% • %■ ■■U - ‘' . ) T ( l i ) - T ( J k * > i )

and noting that (**)!,

we get

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7

I T (/*t'k + i)

_.

TU*H) M ( T C kiT fa+ M

T( » + U T U * * ) T W T f a D T O * *■**•)

Making the above substitution in (1*13) and by making use of (1*12), we have

(1.14)J (x) =

V Ji)* (

Qm c ^O do «

o

& /y

When n 2 0 the series, OO

jtovy)i/nO C&M

L

and the terms here are independent of 0.

The first series

can then be integrated termwise with respect to S over the in­ terval (0,70 • In other words, the integral sign in (1.14) can be placed before or after the summation sign.

J

/y);

J^

{XJ

Therefore,

(*)___ Cxu^^G S " r ( * ) T 0* + i ) )

0

*— *k--o

and, since

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-A 6

8

M '*?MS..* A

(UXL M =

A- o

then

(-')*(Kilt™

E

A- o

fl)

-

c/rd (x YTtm C#o. 0) f

b *)!

and /X»* (i.i5)

L^fx)-

f f i

I x w * M0 CM ( K V ^ < M O ) & e . T( yt)T(M^)J wo

We thus have an integral representation for Jn>Ia(x) for n 2 0 and m real. Recursion Formulas Differentiating with respect to x, the equation,

Immm A- o

A I V (fl> +A

+I V *

becomes

(1.16)

T/n /m M

^ ^ +J* ^ ^ X * * **- • 4 — A zo

lM *lJkJ { T ( m * * + ‘)

Multiplying through by x and separating the series into two stuns, gives

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9

L

oo

rt+xA

I

Jk( T( / n+A* >)

A*i

A-o

s # J/rt m M

f

*

m aA (t ) (fc-l)!71(/nt& +0

* ^ t!i A±!.t " ”)* & ) m -L L - ( A - i) ! T U < A + >)

.

A-l

Replacing the index k by k + 1, the above becomes

* C

M

W ' -

A*— A I T ( / n * & + t )

and by taking (l+/m)X

a

out of the summation, we have

(1.17)

x ai(x) of any order in terms of the functions Jn,m(x) and Jn-l,m(x)

lower

orders* By multiplying equation (1.13) by xn“l and equation (1*17) by x”n“l, we can write these formulas, respectively, as

£ [ x 'M I ,

m

(x)]

1 1 .,/m

W

•

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11 The PunetIons of Integral Orders When n is a negative integer, define

(1 .20 ) A~o

■

-$•T7(-/» +A +1)

Now, if we define l/T(p) to be zero when p = 0, -1, -2, -3, ..., formula (1.9) can be used to define a function Jn,m(x) even when n is a negative integer.

For if n = -j, where j is a

positive integer, that formula becomes

Summing with respect to h, where h = - j+ k, the above can be written

J,- m cx) = yif T " A -9

or since T (h + 1) = hi and ( h + j ) l = T ( h t j + l ) ,

I j m (*) ' (rift f

ft

( i) i— Cmm 4 ! V ( i +J(+i) 4 -o 7

S

.

But the last series represents Jj,m(x)» hence for functions of integral order (i.2i)

z h ftC u m ft Tjj / m ( x) .

( j = 1,1, i , ...)

According to solution (1.20), the function

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12

Is a solution of equation (1.1) and, since (-l)n(l + m)n is In­ dependent of x, y r

a solution when n is a negative

integer; hence the function defined by equation (1.8) is a solution for every real n. When n is not an integer, n and -n do not differ by a whole number except when n is a fraction with the denominator 2; therefore, from the theory of differential equations (Frobenius), Jn,m(x) an^ J-n,m(x) are independent solutions. When n is a fraction with a denominator of 2, we will show later that J_n,m(x) an