# Gamma Function

**Gamma function** is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.

**Gamma function** denoted by is defined as:

where p>0.

Gamma function is also known as Euler’s integral of second kind.

Integrating Gamma function by parts we get,

Thus

**Some standard results:**

We know that

Put t=u^2

Thus

Now changing to polar coordinates by using u = r cosθ and v = r sinθ

Thus

Hence

Where n is a positive integer and m>-1

Put x=e^-y such that dx=-e^{-y}dy=-x dy

Put (m+1)y = u

**Example-1:**

Compute

**Explanation :**

Using

We know

Thus

**Example-2:**

Evaluate

**Explanation :**

Put x^{4} = t, 4x^{3}dx = dt, dx = ¼ t^{-3/4} dt