The Factorial Function - Eulers Gamma

Function

quantic-bolt

January 17th, 2016

1 Purpose

A quick derivation of Eulers gamma function (also known as the factorial func-tion).

2 Proof of Eulers Gamma Function

Eulers gamma function is given by:

(x) =

0

zx1ezdz

Its importance lies in its ability to reproduce the factorial expansion of (n1)!,which lends its usefulness to many-body problems in physics and engineering.A derivation of the factorial expansion from the gamma function via integrationby parts is given below, where u = zx1, du = (x 1)zx2dz, dv = ezdz, andv = ez.

(x) =

0

zx1ezdz (1)

0

zx1ezdz = zx1ez]0 0

(x 1)zx2ezdz (2) 0

zx1ezdz = (x 1) 0

zx2ezdz =(x 1)x 1 (3)

(x) = (x 1)(x 1) (4)Continuing with this trend then, (x 1) = (x 2)(x 2), (x 2) =(x 3)(x 3), and (x n) = (x n + 1)(x n + 1). Therefore:

(x) = (x 1)(x 2)(x 3)...(x n)

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