# 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 [1]).

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 [3]). The Schwarz integral and its generalizations are very important when solving boundary value problems of analytic function theory (see also [3]) and when studying boundary properties of analytic functions (see also [4]).

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 [3], and also Hilbert singular integral).

#### References

 [1] H.A. Schwarz, "Gesamm. math. Abhandl." , 2 , Springer (1890) [2] A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian) [3] F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) [4] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

The Schwarz problem is closely related to the Dirichlet problem: Given the real part $u( t)$ of the boundary value of $f( z)$, the harmonic function $u( x, y)$ is found from it and then the conjugate harmonic function $v( x, y)$ is determined from $u( x, y)$ via the Cauchy-Riemann equations; cf. [3], Sect. 27.2.