4 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS

1.2. A meromorphic extension of the Gamma function

Let a be a complex number, the real part of which satisfies Re(a) −1, and

let φ be a Schwartz function on

R+.

Exercise 1.11. Show that Fa(φ) = −

Fa+1(φ )

a+1

.

After integrating by parts k times we get

(1.6) Fa(φ) =

(−1)k

(a + 1) · · · (a + k)

Fa+k(φ(k)).

Since

φ(k)

is a Schwartz function, the expression on the right-hand side is

well defined provided a + k has its real part larger than −1. The expression

(−1)k

(a+1)···(a+k)

Fa+k(φ(k))

therefore defines an extension of Fa to the half-plane Re(a)

−k − 1. Given a complex number a, there is a positive integer k such that

Re(a) −k − 1, and we set

(1.7)

˜

F

a

(φ) :=

(−1)k

(a + 1) · · · (a + k)

Fa+k(φ(k)).

Exercise 1.12. Show that this definition does not depend on the choice of

k −Re(a) − 1 in checking that

˜

F a(φ) =

(−1)k+l

(a+1)···(a+k+l)

˜

F

a+k+l(φ(k+l))

for any

positive integer l.

Equation (1.7) therefore extends Fa(φ) to a meromorphic function a →

˜

F

a

(φ)

on the plane with simple poles at negative integers. The residue at a negative

integer −k ∈ −N is given by

Resa=−k

˜

F a(φ) = lim

a→−k

(a + k)

˜

F a(φ)

=

(−1)k

(−k + 1) · · · (−1)

∞

0

φ(k)(x)

dx

=

φ(k−1)(0)

(k − 1)!

, (1.8)

where for a meromorphic function f with a simple pole at z0, we have set

Resz=z0 f := lim

z→z0

((z − z0)f(z)) .

When applied to φ : x →

e−x,

this construction provides an extension of the

Gamma function to the whole complex plane, defined on the half-plane Re(b) −k

with k ∈ N by

(1.9)

˜

Γ( b) :=

1

b(b + 1) · · · (b + k − 1)

Γ(b + k).

Exercise 1.13. Show that

˜

Γ has simple poles at integers −k ∈ −N ∪ {0} with

residue at these poles given by

(1.10) Resb=−k

˜

Γ( b) =

(−1)k

k!

.

From now on we use the same notation Γ for the extension

˜

Γ. By (1.9) we have

the following recursive formula:

(1.11) Γ(b + k) = b(b + 1) · · · (b + k − 1) Γ(b) if Re(b) 0.