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
- <math>
\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt
</math>
converges absolutely. Using
integration by parts, one can show that
- <math>\Gamma(z+1)=z\Gamma(z)\,.</math>
Because of Γ(1) = 1, this relation implies
- <math>\Gamma(n+1) = n!\,</math>
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
- <math>\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}.</math>
The Gamma function has a
pole of order 1 at
z=-
n for every
natural number n; the
residue there is given by
- <math>\operatorname{Res}(\Gamma,-n) = \frac{(-1)^n}{n!}</math>
The following multiplicative form of the Gamma function is valid for all complex numbers z which are not non-positive integers:
- <math>\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}</math>
Here γ is the
Euler-Mascheroni constant.