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