# 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

Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the **CS Theory Course** at a student-friendly price and become industry ready.

**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