Schwarz formula

In mathematics, especially complex analysis, the Schwarz formula says: if a complex-valued function $f$ is continuous on the disk and analytic inside, then::$f(z)\; =\; \{1\; over\; 2pi\}\; int\_0^\{2pi\}u(e^\{ipsi\})\; \{e^\{ipsi\}\; +\; z\; over\; e^\{ipsi\}\; -\; z\}dpsi\; +\; iv(0)$ for where we set $f\; =\; u\; +\; iv$ with real-valued functions $u,\; v$.

The formula follows from Poisson formula applied to $u$: [cite web
url=http://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&ots=FTpLISInOP&dq=Schwarz+formula&sig=tYdkW2Mq4IJg-gTIDWVCEI4HKCE
title=Lectures on Entire Functions - Google Book Search
publisher=books.google.com
accessdate=2008-06-26
last=
first=] [The derivation without an appeal to the Poisson formula can be found at: http://planetmath.org/encyclopedia/PoissonFormula.html] $u(z)\; =\; int\_0^\{2pi\}\; u(e^\{ipsi\})\; operatorname\{Re\}\; \{e^\{ipsi\}\; +\; z\; over\; e^\{ipsi\}\; -\; z\}\; dpsi$ for

By means of conformal maps, the formula can be generalized to any simply connected open set.

Notes

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