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)

Comments

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.