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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:¹²

u ( z ) = 1 2 π ∫ 0 2 π u ( e i ψ ) Re ⁡ e i ψ + z e i ψ − z d ψ for  | 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 \qquad {\text{for }}|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

References

1. Lectures on Entire Functions, p. 9, at Google Books https://books.google.com/books?id=NVrgftOGG1sC&pg=PA9&dq=%22Schwarz+formula%22 ↩

2. The derivation without an appeal to the Poisson formula can be found at: https://planetmath.org/schwarzandpoissonformulas Archived 2021-12-24 at the Wayback Machine https://planetmath.org/schwarzandpoissonformulas ↩