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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

<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.

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