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.
Let f be a function holomorphic on the closed unit disc {z ∈ C | |z| ≤ 1}.
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.
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.
The formula follows from Poisson integral formula applied to u:[1][2] By means of conformal maps, the formula can be generalized to any simply connected open set.