Editing Gamma function (section)

=== Relation to other functions ===
* In the first integral defining the gamma function, the limits of integration are fixed. The upper [[incomplete gamma function]] is obtained by allowing the lower limit of integration to vary:<math display="block">\Gamma(z,x) = \int_x^\infty t^{z-1} e^{-t} dt.</math>There is a similar lower incomplete gamma function.
* The gamma function is related to Euler's [[beta function]] by the formula <math display="block">\Beta(z_1,z_2) = \int_0^1 t^{z_1-1}(1-t)^{z_2-1}\,dt = \frac{\Gamma(z_1)\,\Gamma(z_2)}{\Gamma(z_1+z_2)}.</math>
* The [[logarithmic derivative]] of the gamma function is called the [[digamma function]]; higher derivatives are the [[polygamma function]]s.
* The analog of the gamma function over a [[finite field]] or a [[finite ring]] is the [[Gaussian sum]]s, a type of [[exponential sum]].
* The [[reciprocal gamma function]] is an [[entire function]] and has been studied as a specific topic.
* The gamma function also shows up in an important relation with the [[Riemann zeta function]], <math>\zeta (z)</math>. <math display="block">\pi^{-\frac{z}{2}} \; \Gamma\left(\frac{z}{2}\right) \zeta(z) = \pi^{-\frac{1-z}{2}} \; \Gamma\left(\frac{1-z}{2}\right) \; \zeta(1-z).</math> It also appears in the following formula: <math display="block">\zeta(z) \Gamma(z) = \int_0^\infty \frac{u^{z}}{e^u - 1} \, \frac{du}{u},</math> which is valid only for <math>\Re (z) > 1</math>.{{pb}} The logarithm of the gamma function satisfies the following formula due to Lerch: <math display="block">\operatorname{log\Gamma}(z) = \zeta_H'(0,z) - \zeta'(0),</math> where <math>\zeta_H</math> is the [[Hurwitz zeta function]], <math>\zeta</math> is the Riemann zeta function and the prime ({{math|′}}) denotes differentiation in the first variable.
* The gamma function is related to the [[stretched exponential function]]. For instance, the moments of that function are <math display="block">\langle\tau^n\rangle \equiv \int_0^\infty t^{n-1}\, e^{ - \left( \frac{t}{\tau} \right)^\beta} \, \mathrm{d}t = \frac{\tau^n}{\beta}\Gamma \left({n \over \beta }\right).</math>