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A parameter-dependent integral that gives a solution to the Schwarz problem on expressing an [[Analytic function|analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835401.png" /> in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835402.png" /> by the boundary values of its real (or imaginary) part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835403.png" /> on the boundary circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835404.png" /> (see [[#References|[1]]]).
+
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Let on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835405.png" /> a continuous real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835406.png" /> be given. Then the Schwarz integral formulas defining an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835407.png" />, the boundary values of whose real part coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835408.png" /> (or the boundary values of whose imaginary part coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s0835409.png" />), have the form
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
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]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354011.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354012.png" /></td> </tr></table>
+
$$ \tag{* }
 +
f( z)  = Su( z)  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _ { C } u( t)
 +
\frac{t+z}{t- z }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354013.png" /></td> </tr></table>
+
\frac{dt}{t}
 +
+ ic =
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354017.png" /> are arbitrary real constants. The Schwarz integral (*) is closely connected with the [[Poisson integral|Poisson integral]]. The expression
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354018.png" /></td> </tr></table>
+
\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,
 +
$$
  
is often called the Schwarz kernel, and the integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354019.png" /> 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]]]).
+
$$
 +
f( z)
 +
\frac{1}{2 \pi }
 +
\int\limits _ { C } v( t)  
 +
\frac{t+ z}{t-z}
 +
 +
\frac{dt}{t}
 +
+ c _ {1\ } =
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354020.png" /> and of the complete function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354021.png" /> by the given boundary values of the real part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354022.png" /> (or of expressing the boundary values of the real part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354023.png" /> and those of the complete function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354024.png" /> by the given boundary values of the imaginary part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354025.png" />). If the given functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354027.png" /> satisfy a Hölder condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354028.png" />, then the corresponding boundary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354029.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354030.png" /> are expressed by the Hilbert formulas
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354031.png" /></td> </tr></table>
+
\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} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354032.png" /></td> </tr></table>
+
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]]).
Line 27: Line 116:
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354033.png" /> of the boundary value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354034.png" />, the [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354035.png" /> is found from it and then the conjugate harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354036.png" /> is determined from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083540/s08354037.png" /> via the [[Cauchy-Riemann equations]]; cf. [[#References|[3]]], Sect. 27.2.
+
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.

How to Cite This Entry:
Schwarz integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_integral&oldid=31192
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article