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analytic_function [2014/02/21 11:46] nikolaj |
analytic_function [2014/02/25 10:25] nikolaj |
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| == Cauchy's integral formula == | == Cauchy's integral formula == | ||
| $\frac{1}{n!}f^{(n)}(p) = \frac{1}{2\pi\, i} \oint_\gamma \frac{f(z)}{(z-p)^{n+1}}\, \mathrm dz$ | $\frac{1}{n!}f^{(n)}(p) = \frac{1}{2\pi\, i} \oint_\gamma \frac{f(z)}{(z-p)^{n+1}}\, \mathrm dz$ | ||
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| + | Roughly, the Laplace transform uses this for a re-encoding of a functions $f:\mathbb R^+\to\mathbb R$ with Taylor expansion $f(t)=\sum_{n=0}^\infty a_n t^n$, namely by mapping $t^n$ to $s^{-n}\cdot \frac{1}{s}$. | ||
| === Reference === | === Reference === | ||
