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180 9. Elliptic and Hypergeometric Functions

(n) = (n − 1)!.

Hence
(s) is a generalization or analytic continuation of the factorial. The ana-
lytic function
(s) defined for Re(s)> 0 can be continued as a meromorphic func-
tion in the entire plane C with poles on the set −N of negatives of natural numbers.
Using (5.2), we derive the relation

(s + n)

(s) =
s(s + 1)(s + 2) ··· (s + n − 1)
for Re(s)> 0, but the right-hand side is a meromorphic function for Re(s)> −n,
and it defines a meromorphic continuation of
(s) to the half plane Re(s)> −n.
This works on all half planes Re(s)> −n and hence on all of C.
2
(5.3) Remark. By change of variable x = t , dx = 2tdt,and dx/x = 2(dt/t),we
have the following integral formula for the gamma function:

∞ dt
(
2s −1

(s) = 2 t e .
0 t
(5.4) Proposition. For a, b > 0 we have
( π/2
(a)
(b)
2a−1 2b−1
2 cos θ · sin θ dθ = .
0
(a + b)
Proof. We calculate
∞ ∞ " #
( (
2
η

(a)
(b) = 4 ξ 2a−1 2b−1 exp − ξ + η 2 dξ dη
0 0
∞ 2 dr
( ( π/2
e
= 2 r 2a+2b −r · 2 cos 2a−1 θ sin 2b−1 θ dθ
0 r 0
using the transformation dξ dη = rdrdθ to polar coordinates. This proves the
proposition.
π/2
As a special case a = b = 1/2, we have 2 dθ =
(1/2)
(1/2)/
(1) and
√ 0
hence
(1/2) = π. The gamma function prolongs the factorial, and the binomial
coefficient is prolonged by the formula
s s(s − 1) ··· (s − n + 1)

= .
n n!
The corresponding binomial series is

s
s n
(1 + x) = x
n
0≤n
which converges for |x| < 1and Re(s)> 0.