User:Stendhalconques

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[edit] Examples of asymptotic expansions

\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)
xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \rightarrow \infty)
\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 B2m 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 | .

\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
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