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.
