# 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:

1. We know that
Put t=u^2
Thus

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

Hence

2. Where n is a positive integer and m>-1
Put x=e^-y such that dx=-e-ydy=-x dy

Put (m+1)y = u

Example-1:
Compute

Explanation :
Using

We know
Thus

Example-2:
Evaluate

Explanation :
Put x4 = t, 4x3dx = dt, dx = ¼ t-3/4 dt

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