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 .