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's integral formula

Further information: Euler's integral formula for gamma function

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

• Euler's integral formula for gamma function: This states that on the right half-plane:

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

• Recurrence relation for gamma function: This states that if $z$ is not a non-positive integer, we have:

$\Gamma (z+1)=z\Gamma (z)$

• Reflection principle for gamma function: If $z$ is not an integer, we have:

$\Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin(\pi z)}}$

• Integral formula for reciprocal of gamma function
• Reciprocal of gamma function is entire: In other words, the gamma function is zero-free
• Summation formula for logarithmic derivative of gamma function: This states that:

${\frac {\Gamma '(z)}{\Gamma (z)}}=\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+z}}\right)$

• Product formula for reciprocal of gamma function
• Gauss's formula for gamma function

External links

• Template:Mathworld