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:
Gamma function is also known as Euler’s integral of second kind.
Integrating Gamma function by parts we get,
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Some standard results:
We know that
Now changing to polar coordinates by using u = r cosθ and v = r sinθ
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
Put x4 = t, 4x3dx = dt, dx = ¼ t-3/4 dt