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

:$f(z) = \frac \oint_ \frac \operatorname(f(\zeta)) \, \frac+ i\operatorname(f(0))$

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, , ''z''^''α'' ''f''(''z''), is bounded on the closed upper half-plane. Then

:$f(z) = \frac \int_^\infty \frac \, d\zeta = \frac \int_^\infty \frac \, d\zeta$

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'':The derivation without an appeal to the Poisson formula can be found at: https://planetmath.org/schwarzandpoissonformulas

:$u(z) = \frac\int_0^ u(e^) \operatorname \, d\psi \qquad \text , z, < 1.$

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, {{isbn, 0-13-327461-6

Theorems in complex analysis