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:

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

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

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

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

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

$\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)$

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

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

• 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:

$\frac{{\mathrm{\Gamma }}^{\prime }(z)}{\mathrm{\Gamma }(z)}=\sum _{n=0}^{\mathrm{\infty }}(\frac{1}{n+1}-\frac{1}{n+z})$

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

External links

• Template:Mathworld