## Encyclopedia > Gamma function

Article Content

# Gamma function

In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. The notation was thought of by Adrien-Marie Legendre. If the real part of the complex number z is positive, then the integral

$\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt$ converges absolutely. Using integration by parts, one can show that
$\Gamma(z+1)=z\Gamma(z)\,.$

Because of Γ(1) = 1, this relation implies

$\Gamma(n+1) = n!\,$
for all natural numbers n. It can further be used to extend Γ(z) to a holomorphic function defined for all complex numbers z except z = 0, -1, -2, -3, ... by analytic continuation. It is this extended version that is commonly referred to as the Gamma function.

The Gamma function does not have any zeros. Perhaps the most well-known value of the Gamma function at a non-integer is

$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.$
The Gamma function has a pole of order 1 at z=-n for every natural number n; the residue there is given by
$\operatorname{Res}(\Gamma,-n) = \frac{(-1)^n}{n!}$

The following multiplicative form of the Gamma function is valid for all complex numbers z which are not non-positive integers:

$\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}$
Here γ is the Euler-Mascheroni constant.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Autocracy ...     Contents Autocracy Autocracy is a form of government which resides in the absolute power of a single individual. The term can b ...