complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...

an imaginary constant, from the boundary values of its real part.

Unit disc

Let ''f'' be a function holomorphic on the closed unit disc . Then :

f(z) = \frac \oint_ \frac \operatorname(f(\zeta)) \, \frac+ i\operatorname(f(0))

for all , ''z'', < 1.

Upper half-plane

Let ''f'' be a function holomorphic on the closed upper half-plane such that, for some ''α'' > 0, , ''z''^''α'' ''f''(''z''), is bounded on the closed upper half-plane. Then :

f(z) = \frac \int_^\infty \frac \, d\zeta = \frac \int_^\infty \frac \, 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'':The derivation without an appeal to the Poisson formula can be found at: https://planetmath.org/schwarzandpoissonformulas :

u(z) = \frac\int_0^ u(e^) \operatorname  \, d\psi \qquad \text , 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, * Remmert, Reinhold (1990), ''Theory of Complex Functions'', Second Edition, Springer, * 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 Theorems in complex analysis