1550251515-Classical_Complex_Analysis__Gonzalez_

640 Chapter^8

Because of the symmetry of the hypergeometric function with respect

to a and b, we also have

F(a,b,c;z)= r(c) 1

1

ta-^1 (1-w-a-^1 (l-tz)-bdt (8.22-10)

r(a)r(c - a) 0

valid for Rec > Re a > 0, z E D = <C - [1, +oo ). The integrals in (8.22-9)

and (8.22-10) reduce to the integral of the B-function if a = 0 or b = 0. Of

course, :for those values of the parameters the representation is no longer

valid.

(7) Since every term of the series

( )

~ (a)n(b)n

F a,b,c;l = 1+ ~ n!(c)n (8.22-11)

is analytic in a, b, and c separately, it suffices to show that the series

converges uniformly over the region Re( c-a-b) 2:: 2'5, i5 being an arbitrary

positive number. We have, taking into account Theorem 8.48-11.

lim I (a)n(b)nnlH I

n--+oo n!( c )n

r I (a)n+i (b)n+l n!nc (c + n)n

1

H I

= n:_.~ n!na n!nb (c)n+l (a+n)(b+n)nc-a-b

= I r(~j~(b) I J~n;o I (a~): n) I J I nc-a

1

-b-6 I= O

since Re( c - a - b-S) ~ 2'5 - i5 > 0. Hence, for n large enough, say, n ~ N,

I

(a)n(b)n I< 1

n!(c)n nlH

and since L:~=N 1/n^1 H converges, by the Weierstrass M-test the se-

ries (8.22-11) converges absolutely and uniformly on Re(c - a - b) ~

2'5.

(8) We have seen that the hypergeometric series (8.22-11) converges if

Re(c - a - b) > 0. Hence, by Theorem 8.15 (Abel's limit theorem), we

have, with z = x real,

F( a, b, c; 1) = lim F( a, b, c; x)

x--+1-

whenever Re(c - a - b) > 0. If, in addition, the condition Rec> Reb > O

is imposed, we obtain from (8.22-9),

F(a,b,c;l)= lim r(c) f\b-^1 (1-ty-b-^1 (1-xt)-adt

x-1-r(b)r(c - b) lo