Difference between revisions of "Schwarz integral"
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− | + | A parameter-dependent integral that gives a solution to the Schwarz problem on expressing an [[Analytic function|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 [[#References|[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|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 [[#References|[3]]]). The Schwarz integral and its generalizations are very important when solving [[Boundary value problems of analytic function theory|boundary value problems of analytic function theory]] (see also [[#References|[3]]]) and when studying [[Boundary properties of analytic functions|boundary properties of analytic functions]] (see also [[#References|[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 [[#References|[3]]], and also [[Hilbert singular integral|Hilbert singular integral]]). | here the integrals in these formulas are singular integrals and exist in the Cauchy principal-value sense (see [[#References|[3]]], and also [[Hilbert singular integral|Hilbert singular integral]]). | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.A. Schwarz, "Gesamm. math. Abhandl." , '''2''' , Springer (1890)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.A. Schwarz, "Gesamm. math. Abhandl." , '''2''' , Springer (1890)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | The Schwarz problem is closely related to the [[Dirichlet problem|Dirichlet problem]]: Given the real part | + | The Schwarz problem is closely related to the [[Dirichlet problem|Dirichlet problem]]: Given the real part $ u( t) $ |
+ | of the boundary value of $ f( z) $, | ||
+ | the [[Harmonic function|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. [[#References|[3]]], Sect. 27.2. |
Latest revision as of 18:20, 26 January 2022
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.
Schwarz integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_integral&oldid=31192