Schwarz integral formula

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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.

Contents

Unit disc

Let f be a function holomorphic on the closed unit disc {z  C | |z|  1}. Then

for all |z| < 1.

Upper half-plane

Let f be a function holomorphic on the closed upper half-plane {z  C | Im(z)  0} such that, for some α > 0, |zα f(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 , p. 9, at Google Books
  2. The derivation without an appeal to the Poisson formula can be found at: https://planetmath.org/schwarzandpoissonformulas Archived 2021-12-24 at the Wayback Machine

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