Sequences, Series, and Special Functions 639provided that Rec > Re b > 0. Using (8.22-7) in (8.22-2), we obtain
F(a, b, c; z) = r(c) f: (a)nzn 11 tb+n-1(1-w-b-1 dt
r(b)r(c - b) n=O n! o= r(c) f: fl tb-1(1-ty-b-1 (a)n(zt)n dt (8.22-8)
r(b)r(c - b) n=O lo n!
For lzl < 1 the series z::= 0 (a)n(zt)n/nl converges uniformly with re-
spect to t over the interval of integration [O, l]. Hence the order of
summation and integration in (8.22-8) can be reversed, so thatF(a, b, c; z) = r(c) fl tb-1(1-w-b-l f: (a)n(zt)n dt
r(b)r(c - b) lo n=O n!= r(c) f
1
tb-^1 (1-w-b-^1 (1-tz)-a dt (8.22-9)
r(b)r(c - b) lo
The last integral in (8.22-9) is uniformly convergent" with respect to z on
any compact subset of the z-plane cut along [1, +oo ), and so represents
an analytic function of z in C - [1, +oo ). Thus the integral representa-
tion (8.22-9), although established under the assumption lzl < 1, furnishes
an extension of the hypergeometric function to the domain D = C-[1, +oo)
in the case Rec > Re b > 0. In fact, suppose that z belongs to the closed
and bounded set
S={z: 8~ lz-ll~R,Arg(z-l)l~7r-e}
where 8 > 0, e > 0 are arbitrary small, and R > 0 can be taken arbitrarily
large. Then the integrand
g(t, z) = tb-^1 (1 - t)c-b-^1 (1 - tz)-a
where (1 - tz)-a is given its principal value, is a continuous function of
t for every z E Sand analytic in z for every t E [O, 1], and so lg(t,z)I is
bounded in [O, 1] x S. Also, if we let
we have
M = max 1(1-tz)-al
tE[0,1]
zES11 lg(t,z)dt ~ M 11 tReb-1(1-t)Rec-Reb-1 dt
which shows that J 01 g( t, z) dt converges uniformly with respect to z E S