User:Stendhalconques~enwiki

Examples of asymptotic expansions

• Gamma function

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

• Exponential integral

$x{e}^x{E}_1(x)\sim \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n}n!}{x^n}(x\to \mathrm{\infty })$

• Riemann zeta function

$\zeta (s)\sim \sum _{n=1}^{N-1}{n}^{-s}+\frac{N^{1-s}}{s-1}+N^{-s}\sum _{m=1}^{\mathrm{\infty }}\frac{B_{2m}{s}^{\overline{2m-1}}}{(2m)!N^{2m-1}}$

where $B_{2m}$ are Bernoulli numbers and $s^{\overline{2m-1}}$ 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 $N>|s|$.

• Error function

$\sqrt{\pi }x{e}^{x^2}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)=1+\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{(2n)!}{n!(2x{)}^{2n}}.$

== stochastic integral of a process ==.

$\underset{a}{\overset{b}{\int }}{X}_{t}d{B}_t$

corresponding sums of the form

$\sum X_{t_i}(B_{t_{i+1}}-B_{t_i}).$

Itô 's lemma

$dx(t)=a(x,t)dt+b(x,t)d{W}_t$

and let f be some function with a second derivative that is continuous.

Then:

$f(x(t),t)$ is also an Itō process.

$df(x(t),t)=(a(x,t)\frac{\partial f}{\partial x}+\frac{\partial f}{\partial t}+\frac{b(x,t)*b(x,t)*\frac{{\partial }^{2}f}{\partial x^2}}{2})dt+b(x,t)\frac{\partial f}{dx}d{W}_t$