2 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS
For a complex number a with real part Re(a) larger than −1, we can therefore
consider the distribution
Fa :
S(R+)
−→ C
φ −→

0
xa
φ(x) dx.
Fa defines a homogeneous distribution of degree a in the following sense. Given
a Schwartz function f in
S(R+)
and any positive number t, we set φt =
t−1φ(t−1·).
Then,
(1.1) Fa(φt) =
ta
Fa(φ) if Re(a) −1.
The integral
Γ(b) :=

0
xb−1 e−x
dx,
which is defined for Re(b) 0, is called the Gamma function.
The Gamma function z Γ(z) defines a holomorphic map on the half-plane
Re(z) 0.
Exercise 1.4. Check this assertion.
Integration by parts shows that Γ(b) :=
Γ(b+1)
b
for Re(b) 0.
Exercise 1.5. For any complex number b with positive real part show that:
(1.2) Γ(b) :=
1
b(b + 1) · · · (b + k 1)
Γ(b + k) ∀k N.
Deduce that Γ(k) = (k 1)! ∀k N.
Extending the Gamma function to the whole complex plane is related to the
problem of extending homogeneous distributions Fa considered by Hadamard and
Riesz (see e.g. [Sch, Chapter II]) to all complex values a. One wants to assign to
the poles of Γ a finite value, which amounts to assigning a finite value
˜
F −k(φ) to
negative integers −k.
Exercise 1.6. Show that the map x (log x)
e−x
lies in
L1(R+).
Hence we can define the Euler’s
constant1
γ :=

0
log x
e−x
dx.
The following elementary properties of the Gamma function are useful for forth-
coming applications.
Proposition 1.7. (1) The Gamma function is differentiable at any pos-
itive integer k and
(1.3) Γ (1) = −γ;
Γ (k)
Γ(k)
=


k−1
j=1
1
j

γ⎠

∀k N {1}.
1The
Gamma constant was first introduced by Euler in 1735 as the limit γ =
limN→∞

N
n=1
1
n

N
1
1
t
dt .
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