1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 641

= r( c) r1 th-1(1 - t)c-a-b-1 dt

r(b)r(c-b) } 0

=

r(c) r(b)r(c - a - b) r(c)r(c - a - b)

r(b)r(c-b) r(c-a) = r(c-a)r(c-b)

The passage to the limit under the integral sign is justified by the uniform

convergence of the integral over the integral 0 :::; x :::; 1. In fact, since

1-t:::; 1-xt:::; 1

for 0 :::; x :::; 1, 0 :::; t ::=; 1, we have

itb-1(l -w-b-1(l - xt)-al:::; tReb-1(1-t)h-1

where we let h = Re( c - b - a) > 0 if Re a > 0, and h = Re( c - b) > 0 in

the case Re a < 0. Now the B-integral

11 tRe b-1 (1 - t)h-1 dt

converges under the conditions Re( c - b - a) > 0 and Rec > Re b > 0, so,
under such conditions, the integral

11 tb-1(1 - q-b-1(1 - xt)-a dt

converges absolutely and uniformly with respect to x E (0, 1].

However, the restriction Rec > Re b > 0, which is required by the

method of proof used above, can be removed. This follows from the identity
principle for analytic functions, since both sides of the formula

r(c)r(c - a - b)

F( a, b, c; l) = r( c - a )r( c - b)

are analytic in a, b, and c, provided that c is neither zero nor a negative

integer and Re(c - a - b) > 0.

8.23 The Confluent Hypergeometric Function

Definition 8.9 The function

«I>(a, c; z) = f \a())n zn = lim F (a, b, c; ~)

n=O n. c n b-->oo

(8.23-1)

is known as the confluent hypergeometric function, or Kummer function

(also as the Pochhammer-Barnes confluent hypergeometric function). As

before, a and c are complex parameters with c -=f:. 0, -1, -2, .... The