1550251515-Classical_Complex_Analysis__Gonzalez_

644 Chapters

where the reversion of the order of summation and integration is justified

by the uniform convergence of the series I::::"= 0 (zt)n Jn! with respect to t

over the interval [O, I].

( 4) Assume for the moment that Rec> Re a> 0, so as to apply {8.23-3).

Letting t = 1 - T, this formula yields

«I>(a,c;z)= r{c). ez f-zrTc-a-l(I-rt-ldr

r(a)r(c - a) lo

or

«I>(a, c; z) = ez«I>(c - a, c; -z) {8.23-4)

since c --( c - a) = a and Rec > Re( c - a) > 0.

To remove the restriction imposed in the proof, it suffices to observe that,

after division by r(c), both sides of (8.23-4) become analytic functions of

a and c, provided that c #-O, -1, -2,.... By the identity principle

for analytic functions the two sides are equal for all values of a and all
admissible values of c. The relation {8.23-4) is due to E. E. Kummer [20]
{sometimes called Kummer's first formula).
(5) By letting () = z(d/dz) we get

()(()+c-l)«f>=f n(n+c-l)(a)nzn=f (a)n Zn

n=O n!(c)n n=l (n -l)!(c)n-1

= f (a)n+i zn+l = z f (a+ n)(a)n Zn

n=O n!(c)n n=O n!(c)n

= z(() + a)«I>

so that

[()(() + c -1)-z(() + a)]«I> = 0

which gives

z«I>" + ( c - z )«I>' - a«I> = 0

since ()cp = z«I>' and e(e,-l)«I> = z^2 «I>".

A second linearly independent solution of the differential equation

zw" + ( c - z )w' - aw = 0 {8.23-5)

can be obtained by letting w = z^1 -c(, provided that c #-0, ±1, ±2, ....

When the substitution is made (8.23-5) becomes