Editing Gamma function (section)

== Approximations ==
[[File:Mplwp factorial gamma stirling.svg|thumb|right|upright=1.35|Comparison of the gamma function (blue line) with the factorial (blue dots) and Stirling's approximation (red line)]]
Complex values of the gamma function can be approximated using [[Stirling's approximation]] or the [[Lanczos approximation]], 
<math display="block">\Gamma(z) \sim \sqrt{2\pi}z^{z-1/2}e^{-z}\quad\hbox{as }z\to\infty\hbox{ in } \left|\arg(z)\right|<\pi.</math>
This is precise in the sense that the ratio of the approximation to the true value approaches 1 in the limit as {{math|{{abs|''z''}}}} goes to infinity.

The gamma function can be computed to fixed precision for <math>\operatorname{Re} (z) \in [1, 2]</math> by applying [[integration by parts]] to Euler's integral. For any positive number&nbsp;{{mvar|x}} the gamma function can be written
<math display="block">\begin{align}
\Gamma(z) &= \int_0^x e^{-t} t^z \, \frac{dt}{t} + \int_x^\infty e^{-t} t^z\, \frac{dt}{t} \\
&= x^z e^{-x} \sum_{n=0}^\infty \frac{x^n}{z(z+1) \cdots (z+n)} + \int_x^\infty e^{-t} t^z \, \frac{dt}{t}.
\end{align}</math>

When {{math|Re(''z'') ∈ [1,2]}} and <math>x \geq 1</math>, the absolute value of the last integral is smaller than <math>(x + 1)e^{-x}</math>. By choosing a large enough <math>x</math>, this last expression can be made smaller than <math>2^{-N}</math> for any desired value <math>N</math>. Thus, the gamma function can be evaluated to <math>N</math> bits of precision with the above series.

A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba.<ref>E.A. Karatsuba, Fast evaluation of transcendental functions. Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991).</ref><ref>E.A. Karatsuba, On a new method for fast evaluation of transcendental functions. Russ. Math. Surv. Vol.46, No.2, pp. 246–247 (1991).</ref><ref>E.A. Karatsuba "[http://www.ccas.ru/personal/karatsuba/algen.htm Fast Algorithms and the FEE Method]".</ref>

For arguments that are integer multiples of {{math|{{sfrac|1|24}}}}, the gamma function can also be evaluated quickly using [[arithmetic–geometric mean]] iterations (see [[particular values of the gamma function]]).<ref>{{cite journal |last1=Borwein |first1=J. M. |last2=Zucker |first2=I. J. |title=Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind |journal=IMA Journal of Numerical Analysis |date=1992 |volume=12 |issue=4 |pages=519–526 |doi=10.1093/IMANUM/12.4.519}}</ref>