Gamma function

Definition

The gamma function is a meromorphic function on $\mathbb {C}$, with simple poles at all the non-positive integers, having residue at $-k$ equal to:

${\frac {(-1)^{k}}{k!}}$

It is defined in a number of equivalent ways.

Euler integral formula

This defines the gamma function in the right half-plane by the formula:

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt$

and extended analytically to $\mathbb {C}$ by the prescription:

$\Gamma (z)={\frac {\Gamma (z+n)}{z(z+1)(z+2),\dots ,(z+n-1)}}$

where $n$ is an integer chosen such that $z+n$ has positive real part.

Facts

• Functional equation for gamma function
• Reflection principle for gamma function
• Reciprocal of gamma function is entire

External links

• Template:Mathworld