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analytic_function [2014/02/21 11:46] nikolaj |
analytic_function [2014/03/21 11:11] 127.0.0.1 external edit |
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===== Analytic function ===== | ===== Analytic function ===== | ||
==== Set ==== | ==== Set ==== | ||
- | | @#88DDEE: $\mathcal O\subset \mathbb C$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal O\subset \mathbb C$ | |
- | | @#FFBB00: $f\in \mathrm{it}$ | | + | | @#FFBB00: definiendum | @#FFBB00: $f\in \mathrm{it}$ | |
- | | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | | + | | @#AAFFAA: inclusion | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | |
- | | @#FFFDDD: $c$ ... series in $\mathbb C$ | | + | | @#FFFDDD: for all | @#FFFDDD: $c$ ... series in $\mathbb C$ | |
>todo, roughly | >todo, roughly | ||
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== Cauchy's integral formula == | == Cauchy's integral formula == | ||
$\frac{1}{n!}f^{(n)}(p) = \frac{1}{2\pi\, i} \oint_\gamma \frac{f(z)}{(z-p)^{n+1}}\, \mathrm dz$ | $\frac{1}{n!}f^{(n)}(p) = \frac{1}{2\pi\, i} \oint_\gamma \frac{f(z)}{(z-p)^{n+1}}\, \mathrm dz$ | ||
+ | |||
+ | 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 === |