Schwarz integral formula

In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part.

Unit disc

Let ƒ = u + iv be a function which is holomorphic on the closed unit disc {z  C | |z|  1}. Then

for all |z| < 1.

Upper half-plane

Let ƒ = u + iv be a function that is holomorphic on the closed upper half-plane {z  C | Im(z)  0} such that, for some α > 0, |zα ƒ(z)| is bounded on the closed upper half-plane. Then

for all Im(z) > 0.

Note that, as compared to the version on the unit disc, this formula does not have an arbitrary constant added to the integral; this is because the additional decay condition makes the conditions for this formula more stringent.

Corollary of Poisson integral formula

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

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

Notes and references

  1. "Lectures on Entire Functions - Google Book Search". books.google.com. Retrieved 2008-06-26.
  2. The derivation without an appeal to the Poisson formula can be found at: http://planetmath.org/encyclopedia/PoissonFormula.html
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