Sequences, Series, and Special Functions 615Corollary 8.23 The gamma function can be represented in the form
oo ( l)n
r(z) =I: 1 -. + Q(z), Rez > 0 (8.20-5)n=O n.(z+n)
The series on the right.of (8.20-5) converges for all z f: 0, -1, -2, ...
and since Q(z) is an entire function, the representation (8.20-5) may serve
to extend the definition of r(z) to the whole complex plane excepting thepoints O, -1, -2, .... Formula (8.20-5) is called Prym's decomposition of
the r-function.
Proof We haveP(z) = f
1
e-tt':-^1 dt = f1[f (-lr tn] tz-^1 dt
lo lo n=O n.= f (-l)n {1 tn+z-1 dt = f (-1r
n=O n! lo n=O n!(z + n)(8.20-6)the integration term by term of the series being justified by Corollary 8.1.
Substitution of (8.20-6) in (8.20-2) gives (8.20-5). Let
00
G = n {z: lz + nl ;::: 8}
n=Owhere 8 > 0 is arbitrary. For z E G we have
I
(~l)n I 1
n!(z + n) ~ n!8
and since Z:::'::o n~S converges, the series on the right of (8.20-5) converges
absolutely and uniformly on G; Because of the arbitrariness of 8, the
function
o6 (-l)n
f(z) = f; n!(z + n) + Q(z) (8.20-7)
is analytic in CC with the points 0, -1, -2, ... deleted. Since r(z) =
f(z) for Rez > 0, we may use (8.20-5) to extend the definition of r(z)
to the complex plane, except for the points O, -1, -2,.... As such a
representation suggests, the points 0, -1, -2, ... are poles of the extended
r(z). This will be shown rigorously later (Theorem 8.48-7). If we take this
for granted; we see that r(z) is a meromorphic function in CC.
Theorem 8.47 The integral B(z, () = J 01 tz-^1 (1-t)(-l dt converges ab-