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

[math]\displaystyle{ f(z) = \frac{1}{2\pi i} \oint_{|\zeta| = 1} \frac{\zeta + z}{\zeta - z} \operatorname{Re}(f(\zeta)) \, \frac{d\zeta}{\zeta}+ i\operatorname{Im}(f(0)) }[/math]

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

[math]\displaystyle{ f(z) = \frac{1}{\pi i} \int_{-\infty}^\infty \frac{u(\zeta,0)}{\zeta - z} \, d\zeta = \frac{1}{\pi i} \int_{-\infty}^\infty \frac{\operatorname{Re}(f)(\zeta+0i)}{\zeta - z} \, d\zeta }[/math]

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:

[math]\displaystyle{ u(z) = \frac{1}{2\pi}\int_0^{2\pi} u(e^{i\psi}) \operatorname{Re} {e^{i\psi} + z \over e^{i\psi} - z} \, d\psi \qquad \text{for } |z| \lt 1. }[/math]

This is equivalent to

[math]\displaystyle{ \frac{1}{2\pi} \int u(e^{i\psi}) \frac{cos(2\psi)}{cos(2\psi)-sin(2\psi)+\Re^2(z)+2\Re(z)\Im(z)-\Im^2(z)}-\frac{\Re^2(z)-\Im^2(z)}{\cos(2\psi)-\sin(2\psi)+\Re^2(z)+2\Re(z)\Im(z)-\Im^2(z)} d\psi }[/math]

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

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