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 {z Ã¢ÂˆÂˆ C | |z| Ã¢Â‰Â¤ 1}. Then

for all |z|&nbsp;<&nbsp;1.

Upper half-plane

Let f be a function holomorphic on the closed upper half-plane {z&nbsp;∈&nbsp;C&nbsp;|&nbsp;Im(z)&nbsp;≥&nbsp;0} such that, for some α&nbsp;>&nbsp;0, |z^α&nbsp;f(z)| is bounded on the closed upper half-plane. Then

for all Im(z)&nbsp;>&nbsp;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&nbsp;u:

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

Notes and references

• Ahlfors, Lars V. (1979), Complex Analysis, Third Edition, McGraw-Hill,
• Remmert, Reinhold (1990), Theory of Complex Functions, Second Edition, Springer,
• Saff, E. B., and A. D. Snider (1993), Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Second Edition, Prentice Hall,