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 {z ∈ C | |z| ≤ 1}. Then

$f(z)=\frac{1}{2\pi i}{\oint }_{|\zeta |=1}\frac{\zeta +z}{\zeta -z}\mathrm{Re}(f(\zeta ))\frac{d\zeta }{\zeta }+i\mathrm{Im}(f(0))$

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

$f(z)=\frac{1}{\pi i}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{u(\zeta ,0)}{\zeta -z}d\zeta =\frac{1}{\pi i}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\mathrm{Re}(f)(\zeta +0i)}{\zeta -z}d\zeta$

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.

Corollary of Poisson integral formula

The formula follows from Poisson integral formula applied to u:

$u(z)=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}u(e^{i\psi })\mathrm{Re}\frac{e^{i\psi }+z}{e^{i\psi }-z}d\psi \text{for }|z|<1.$

This is equivalent to

$\frac{1}{2\pi }\int u(e^{i\psi })\frac{cos(2\psi )}{cos(2\psi )-sin(2\psi )+{\mathrm{\Re }}^2(z)+2\mathrm{\Re }(z)\mathrm{\Im }(z)-{\mathrm{\Im }}^2(z)}-\frac{{\mathrm{\Re }}^2(z)-{\mathrm{\Im }}^2(z)}{\cos (2\psi )-\sin (2\psi )+{\mathrm{\Re }}^2(z)+2\mathrm{\Re }(z)\mathrm{\Im }(z)-{\mathrm{\Im }}^2(z)}d\psi$

$=\frac{1}{2\pi }\int u(e^{i\psi })\frac{cos(2\psi )}{cos(2\psi )-sin(2\psi )+{\mathrm{\Re }}^2(z)+2\mathrm{\Re }(z)\mathrm{\Im }(z)-{\mathrm{\Im }}^2(z)}d\psi -[\arctan (\frac{-\tan (x)+{\mathrm{\Re }}^2(z)\tan (x)+2\mathrm{\Re }(z)\mathrm{\Im }(z)\tan (x)-1}{\sqrt{{\mathrm{\Im }}^4(z)-4\mathrm{\Re }(z){\mathrm{\Im }}^3(z)+2{\mathrm{\Re }}^2(z){\mathrm{\Im }}^2(z)+4{\mathrm{\Re }}^3(z)+{\mathrm{\Re }}^4(z)-2}})(\frac{1}{2\pi })+\pi \mathrm{sgn}(2{\mathrm{\Re }}^2(z)+2{\mathrm{\Im }}^2(z)+4\mathrm{\Re }(z)\mathrm{\Im }(z)-2)\lfloor \frac{1}{2}+\frac{x}{\pi }\rfloor \frac{1}{2\pi }]\frac{{\mathrm{\Re }}^2(z)-{\mathrm{\Im }}^2(z)}{\sqrt{{\mathrm{\Im }}^4(z)-4\mathrm{\Re }(z){\mathrm{\Im }}^3(z)+2{\mathrm{\Re }}^2(z){\mathrm{\Im }}^2(z)+4{\mathrm{\Re }}^3(z)+{\mathrm{\Re }}^4(z)-2}}$

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, ISBN 0-07-085008-9
• Remmert, Reinhold (1990), Theory of Complex Functions, Second Edition, Springer, ISBN 0-387-97195-5
• 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