Editing Asymptotic expansion (section)

==Examples==

[[File:AsymptoticExpansionExample.svg|thumb|Plots of the absolute value of the fractional error in the asymptotic expansion of the Gamma function (left). The horizontal axis is the number of terms in the asymptotic expansion. Blue points are for {{nowrap|1=''x''&thinsp;=&thinsp;2}} and red points are for {{nowrap|1=''x''&thinsp;=&thinsp;3}}. It can be seen that the least error is encountered when there are 14 terms for {{nowrap|1=''x''&thinsp;=&thinsp;2}}, and 20 terms for {{nowrap|1=''x''&thinsp;=&thinsp;3}}, beyond which the error diverges.]]

* [[Gamma function]] ([[Stirling's approximation]]) <math display="block"> \frac{e^x}{x^x\sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots\ (x \to \infty)</math>
* [[Exponential integral]] <math display="block">x e^x E_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) </math>
* [[Logarithmic integral]] <math display="block">\operatorname{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k}</math>
* [[Riemann zeta function]] <math display="block">\zeta(s) \sim \sum_{n=1}^{N}n^{-s} + \frac{N^{1-s}}{s-1} - \frac{N^{-s}}{2} + N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^{\overline{2m-1}}}{(2m)! N^{2m-1}}</math>where <math>B_{2m}</math> are [[Bernoulli numbers]] and <math>s^{\overline{2m-1}}</math> is a [[rising factorial]]. This expansion is valid for all complex ''s'' and is often used to compute the zeta function by using a large enough value of ''N'', for instance <math>N > |s|</math>.
* [[Error function]]<math display="block"> \sqrt{\pi}x e^{x^2}{\rm erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{(2x^2)^n} \ (x \to \infty)</math> where {{math|(2''n''&thinsp;−&thinsp;1)!!}} is the [[double factorial]].