# Schwarz integral

A parameter-dependent integral that gives a solution to the Schwarz problem on expressing an analytic function $f( z) = u( z) + iv( z)$ in the unit disc $D$ by the boundary values of its real (or imaginary) part $u$ on the boundary circle $C$( see ).

Let on the unit circle $C = \{ {z } : {z = e ^ {i \phi }, 0< \phi < 2 \pi } \}$ a continuous real-valued function $u( \phi )$ be given. Then the Schwarz integral formulas defining an analytic function $f( z) = u( z) + iv( z)$, the boundary values of whose real part coincide with $u( \phi )$( or the boundary values of whose imaginary part coincide with $v( \phi )$), have the form

$$\tag{* } f( z) = Su( z) = \frac{1}{2 \pi i } \int\limits _ { C } u( t) \frac{t+z}{t- z } \frac{dt}{t} + ic =$$

$$= \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \frac{e ^ {i \phi } + re ^ {i \theta } }{e ^ {i \phi } - re ^ {i \theta } } u( \phi ) d \phi + ic,$$

$$f( z) = \frac{1}{2 \pi } \int\limits _ { C } v( t) \frac{t+ z}{t-z} \frac{dt}{t} + c _ {1\ } =$$

$$= \ \frac{i}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \frac{e ^ {i \phi } + re ^ {i \theta } }{e ^ {i \phi } - re ^ {i \theta } } v( \phi ) d \phi + c _ {1} ,$$

where $z = re ^ {i \theta }$, $t = e ^ {i \phi }$, and $c$ and $c _ {1}$ are arbitrary real constants. The Schwarz integral (*) is closely connected with the Poisson integral. The expression

$$\frac{1}{2 \pi } \frac{e ^ {i \phi } + re ^ {i \theta } }{e ^ {i \phi } - re ^ {i \theta } }$$

is often called the Schwarz kernel, and the integral operator $S$ in the first formula of (*) is called the Schwarz operator. These notions can be generalized to the case of arbitrary domains in the complex plane (see ). The Schwarz integral and its generalizations are very important when solving boundary value problems of analytic function theory (see also ) and when studying boundary properties of analytic functions (see also ).

When applying the integral formulas (*), a very important and more difficult problem arises concerning the existence and the expression of the boundary values of the imaginary part $v( z)$ and of the complete function $f( z)$ by the given boundary values of the real part $u( \phi )$( or of expressing the boundary values of the real part $u( z)$ and those of the complete function $f( z)$ by the given boundary values of the imaginary part $v( \phi )$). If the given functions $u( \phi )$ or $v( \phi )$ satisfy a Hölder condition on $C$, then the corresponding boundary values of $v( \phi )$ or $u( \phi )$ are expressed by the Hilbert formulas

$$v( \phi ) = - \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } u( \alpha ) \mathop{\rm cotan} \frac{ \alpha - \phi }{2} d \alpha + c,$$

$$u( \phi ) = \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } v( \alpha ) \mathop{\rm cotan} \frac{\alpha - \phi }{2} d \alpha + c _ {1} ;$$

here the integrals in these formulas are singular integrals and exist in the Cauchy principal-value sense (see , and also Hilbert singular integral).

How to Cite This Entry:
Schwarz integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_integral&oldid=52001
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article