## The Gamma Function

The gamma function extends the factorial function which acts on non – negative integers to complex numbers.

We consider the improper integrals

We can easily findand can In for n=1,2,3,... by obtaining a reduction formula for

Hence

In generalso for non negative integers the integral exists and takes the values of the factorial function. For general complex numbers the integral is denotedand is called the Gamma function,defined as

Thefunction is analytic on the half planebut analytic continuation can extend the domain to the whole complex plane with the exception of the non – positive integers where it has simple poles.