Editing Incomplete gamma function (section)

===Asymptotic behavior===

* <math>\frac{\gamma(s,x)}{x^s} \to \frac{1}{s}</math> as <math>x \to 0</math>,
* <math>\frac{\Gamma(s,x)}{x^s} \to -\frac{1}{s}</math> as <math>x \to 0</math> and <math>\Re (s) < 0</math> (for real {{math|''s''}}, the error of {{math|Γ(''s'', ''x'') ~ −''x''<sup>''s''</sup> / ''s''}} is on the order of {{math|''O''(''x''<sup>min{''s'' + 1, 0}</sup>)}} if {{math|''s'' ≠ −1}} and {{math|''O''(ln(''x''))}} if {{math|1=''s'' = −1}}),
* <math>\Gamma(s,x) \sim \Gamma(s) - \sum_{n=0}^\infty (-1)^n \frac{x^{s+n}}{n!(s+n)}</math> as an [[asymptotic series]] where <math>x\to0^+</math> and <math>s\neq 0,-1,-2,\dots</math>.<ref name="auto">{{cite book |last=Bender & Orszag |date=1978 |title = Advanced Mathematical Methods for Scientists and Engineers |publisher=Springer|bibcode=1978amms.book.....B }}</ref>
* <math>\Gamma(-N,x) \sim C_N + \frac{(-1)^{N+1}}{N!} \ln x - \sum_{n=0,n\ne N}^\infty (-1)^n \frac{x^{n-N}}{n!(n-N)}</math> as an [[asymptotic series]] where <math>x \to 0^+</math> and <math>N = 1, 2, \dots</math>, where <math display="inline">C_N = \frac{(-1)^{N+1}}{N!} \left( \gamma - \displaystyle\sum_{n=1}^N \frac{1}{n} \right)</math>, where <math>\gamma</math> is the [[Euler-Mascheroni constant]].<ref name="auto"/>
* <math>\gamma(s,x) \to \Gamma(s)</math> as <math>x \to \infty</math>,
* <math>\frac{\Gamma(s,x)}{x^{s-1} e^{-x}} \to 1</math> as <math>x \to \infty</math>,
* <math>\Gamma(s,z) \sim z^{s-1} e^{-z} \sum_{k=0} \frac {\Gamma(s)} {\Gamma(s-k)} z^{-k}</math> as an [[asymptotic series]] where <math>|z| \to \infty</math> and <math>\left|\arg z\right| < \tfrac{3}{2} \pi</math>.<ref>{{Cite web|url=https://dlmf.nist.gov/8.11|title=DLMF: §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions|website=dlmf.nist.gov}}</ref>