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:

$\mathrm{\Gamma }(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{z-1}{e}^{-t}dt$

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

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

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