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 f be a function holomorphic on the closed unit disc. Then
for all |z| < 1.

Upper half-plane

Let f be a function holomorphic on the closed upper half-plane such that, for some α > 0, |zα f| is bounded on the closed upper half-plane. Then
for all Im > 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:
By means of conformal maps, the formula can be generalized to any simply connected open set.