Schwarz formula

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In mathematics, especially complex analysis, the Schwarz formula says: if a complex-valued function f is continuous on the disk | z | < 1 and analytic inside, then:

f(z) = {1 \over 2\pi} \int_0^{2\pi}u(e^{i\psi}) {e^{i\psi} + z \over e^{i\psi} - z}d\psi + iv(0) for | z | < 1

where we set f = u + iv with real-valued functions u,v.

The formula follows from Poisson formula applied to u [1] [2]:

u(z) = \int_0^{2\pi} u(e^{i\psi}) \operatorname{Re} {e^{i\psi} + z \over e^{i\psi} - z} d\psi for | z | < 1

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

[edit] Notes

  1. ^ http://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&ots=FTpLISInOP&dq=Schwarz+formula&sig=tYdkW2Mq4IJg-gTIDWVCEI4HKCE
  2. ^ The derivation without an appeal to the Poisson formula can be found at http://planetmath.org/encyclopedia/PoissonFormula.html


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