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 ) = 1 2 π i ∮ | ζ | = 1 ζ + z ζ − z Re ( f ( ζ ) ) d ζ ζ + i Im ( f ( 0 ) ) {\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))}
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, |zα f(z)| is bounded on the closed upper half-plane. Then
f ( z ) = 1 π i ∫ − ∞ ∞ u ( ζ , 0 ) ζ − z d ζ = 1 π i ∫ − ∞ ∞ Re ( f ) ( ζ ) ζ − z d ζ {\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 )}{\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 ) = 1 2 π ∫ 0 2 π u ( e i ψ ) Re e i ψ + z e i ψ − z d ψ , | z | < 1. {\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 ,\quad |z|<1.}
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
