Gamma function

Gamma
The gamma function along part of the real axis
General information
General definition	${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt}$
Fields of application	Calculus, mathematical analysis, statistics, physics

In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function ${\displaystyle \Gamma (z)}$ is defined for all complex numbers ${\displaystyle z}$ except non-positive integers, and for every positive integer ${\displaystyle z=n}$, $${\displaystyle \Gamma (n)=(n-1)!\,.}$$The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:

$${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}{\text{ d}}t,\ \qquad \Re (z)>0\,.}$$The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles.

The gamma function has no zeros, so the reciprocal gamma function ⁠1/Γ(z)⁠ is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:

$${\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)\,.}$$

Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It appears as a factor in various probability-distribution functions and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics.