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

f ( z ) = 1 2 π i ∮ | ζ | = 1 ζ + z ζ − z Re ( f ( ζ ) ) d ζ ζ + i Im ( f ( 0 ) )

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

f ( z ) = 1 π i ∫ − ∞ ∞ u ( ζ , 0 ) ζ − z d ζ = 1 π i ∫ − ∞ ∞ R e ( f ) ( ζ + 0 i ) ζ − z d ζ

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:

u ( z ) = 1 2 π ∫ 0 2 π u ( e i ψ ) Re ⁡ e i ψ + z e i ψ − z d ψ  for  | z | < 1.

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

References

Schwarz integral formula Wikipedia
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