Complex Analysis/Special functions

Gamma function The Gamma function is the unique function that is meromorphic on and that is given by whenever . Gamma function interpolates the factorial For , we have . We use induction on . The base case is , existence and uniqueness of the Gamma function The integral converges whenever , and there exists a unique function which is meromorphic on and satisfies whenever . 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 .