Asymptotic Expansion - Examples of Asymptotic Expansions

Examples of Asymptotic Expansions

Gamma function

$\frac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots \ (x \rightarrow \infty)$

Exponential integral

Riemann zeta function

$\zeta(s) \sim \sum_{n=1}^{N-1}n^{-s} + \frac{N^{1-s}}{s-1} + N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^{\overline{2m-1}}}{(2m)! N^{2m-1}}$
where are Bernoulli numbers and 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 .