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 ƒ = u + iv be a function which is 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} \text{Re}(f(\zeta)) \, \frac{d\zeta}{\zeta} + i\text{Im}(f(0))

for all |z| < 1.

Upper half-plane

Let ƒ = u + iv be a function that is holomorphic on the closed upper half-plane {z ∈ C | Im(z) ≥ 0} such that, for some α > 0, |z^α ƒ(z)| is bounded on the closed upper half-plane. Then

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{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:^[1]^[2]

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\text{ for }|z| < 1.

By means of conformal maps, the formula can be generalized to any simply connected open set.

Notes and references

↑ "Lectures on Entire Functions - Google Book Search". books.google.com. Retrieved 2008-06-26.
↑ The derivation without an appeal to the Poisson formula can be found at: http://planetmath.org/encyclopedia/PoissonFormula.html

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

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