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 . Then
f(z)=
| 1 | |
| 2\pii |
\oint|\zeta|
| \zeta+z | |
| \zeta-z |
\operatorname{Re}(f(\zeta))
| d\zeta | |
| \zeta |
+i\operatorname{Im}(f(0))
for all |z| < 1.
Let f be a function holomorphic on the closed upper half-plane such that, for some α > 0, |zα f(z)| is bounded on the closed upper half-plane. Then
f(z)=
| 1 | |
| \pii |
| infty | |
| \int | |
| -infty |
| u(\zeta,0) | |
| \zeta-z |
d\zeta=
| 1 | |
| \pii |
| infty | |
| \int | |
| -infty |
| \operatorname{Re | |
| (f)(\zeta+0i)}{\zeta |
-z}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.
The formula follows from Poisson integral formula applied to u:[1]
u(z)=
| 1 | |
| 2\pi |
| 2\pi | |
| \int | |
| 0 |
u(ei\psi)\operatorname{Re}{ei\psi+z\overei\psi-z}d\psi for|z|<1.
By means of conformal maps, the formula can be generalized to any simply connected open set.