Complex Analysis/Special functions

Definition (Gamma function):

The Gamma function is the unique function that is meromorphic on and that is given by

whenever .

Proposition (Gamma function interpolates the factorial):

For , we have


Proof: We use induction on . The base case is ,

Proposition (existence and uniqueness of the Gamma function):

The integral

converges whenever , and there exists a unique function which is meromorphic on and satisfies

whenever .

Proof: First, note that the integral

converges for , because we have the estimate

where is sufficiently large. The first integral evaluates to


whereas the second integral is less than .