User:Stendhalconques~enwiki

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)$$
 * Exponential integral
 * $$xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \  (x \rightarrow \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}^\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}{\rm erfc}(x) = 1+\sum_{n=1}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}.$$

stochastic integral of a process
.


 * $$\int_{a}^{b} X_t\, dB_t$$

corresponding sums of the form


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

Itô 's lemma


 * $$ dx(t) = a(x,t)\,dt + b(x,t)\,dW_t$$

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

Then:
 * $$ f(x(t),t) $$ is also an Itō process.
 * $$ df(x(t),t) = \left( a(x,t)\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b(x,t)*b(x,t)* \frac{\partial^2f}{\partial x^2}}{2} \right) dt + b(x,t)\frac{\partial f}{dx}\,dW_t$$