Editing Gamma function (section)

=== Stirling's formula ===
{{Main|Stirling's approximation}}

[[File:Gamma cplot.svg|thumb|Representation of the gamma function in the complex plane. Each point <math>z</math> is colored according to the argument of {{nowrap|<math>\Gamma(z)</math>.}} The contour plot of the modulus <math>|\Gamma(z)|</math> is also displayed.]]

[[File:Gamma abs 3D.png|thumb|3-dimensional plot of the absolute value of the complex gamma function]]
The behavior of <math>\Gamma(x)</math> for an increasing positive real variable is given by [[Stirling's formula]]
<math display="block">\Gamma(x+1)\sim\sqrt{2\pi x}\left(\frac{x}{e}\right)^x,</math>
where the symbol <math>\sim</math> means asymptotic convergence: the ratio of the two sides converges to 1 in the limit {{nowrap|<math display="inline">x \to + \infty</math>.<ref name="Davis"/>}}  This growth is faster than exponential, <math>\exp(\beta x)</math>, for any fixed value of <math>\beta</math>.

Another useful limit for asymptotic approximations for <math>x \to + \infty</math> is:
<math display="block">  {\Gamma(x+\alpha)}\sim{\Gamma(x)x^\alpha}, \qquad \alpha \in \Complex. </math>

When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function: <ref>{{cite book |last1=Artin |first1=Emil |title=The Gamma Function |date=2015 |page = 24|publisher=Dover }}</ref>
<math display="block"> \Gamma(x) = \sqrt{\frac{2\pi}{x}} \left(\frac{x}{e}\right)^x  \prod_{n=0}^{\infty} \left[\frac{1}{e}\left(1+\frac{1}{x+n}\right)^{x+n+\frac{1}{2}} \right]</math>