Editing Gamma function (section)

{{Short description|Extension of the factorial function}}
{{log(x)}}
{{About||the gamma function of ordinals|Veblen function|the gamma distribution in statistics|Gamma distribution|the function used in video and image color representations|Gamma correction}}
{{Use dmy dates|date=December 2016}}

{{Infobox mathematical function
| name = Gamma
| image = Gamma plot.svg
| imagesize = 325px
| caption = The gamma function along part of the real axis
| general_definition = <math>\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,dt</math>
| fields_of_application = Calculus, mathematical analysis, statistics, physics 
}}

In [[mathematics]], the '''gamma function''' (represented by Γ, capital [[Greek alphabet|Greek]] letter [[gamma]]) is the most common extension of the [[factorial function]] to [[complex number]]s. Derived by [[Daniel Bernoulli]], the gamma function <math>\Gamma(z)</math> is defined for all complex numbers <math>z</math> except non-positive integers, and for every [[positive integer]] <math>z=n</math>, <math display="block">\Gamma(n) = (n-1)!\,.</math>The gamma function can be defined via a convergent [[improper integral]] for complex numbers with positive real part:

<math display="block"> \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\text{ d}t, \ \qquad \Re(z) > 0\,.</math>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 function|holomorphic]] except at zero and the negative integers, where it has simple [[Zeros and poles|poles]].

The gamma function has no zeros, so the [[reciprocal gamma function]] {{math|{{sfrac|1|Γ(''z'')}}}} is an [[entire function]]. In fact, the gamma function corresponds to the [[Mellin transform]] of the negative [[exponential function]]:

<math display="block"> \Gamma(z) =  \mathcal M \{e^{-x} \} (z)\,.</math>

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]].