Let f be a function holomorphic on the closed unit disc {z ∈ C | |z| ≤ 1}. Then
for all |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. 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.
The formula follows from Poisson integral formula applied to u:12
By means of conformal maps, the formula can be generalized to any simply connected open set.
Lectures on Entire Functions, p. 9, at Google Books https://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&dq=%22Schwarz+formula%22 ↩
The derivation without an appeal to the Poisson formula can be found at: https://planetmath.org/schwarzandpoissonformulas Archived 2021-12-24 at the Wayback Machine https://planetmath.org/schwarzandpoissonformulas ↩