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Formulas first discovered by Yu.V. Sokhotskii [[#References|[1]]], expressing the boundary values of a Cauchy-type integral. With more complete proofs but significantly later, the formulas were obtained independently by J. Plemelj .
 
Formulas first discovered by Yu.V. Sokhotskii [[#References|[1]]], expressing the boundary values of a Cauchy-type integral. With more complete proofs but significantly later, the formulas were obtained independently by J. Plemelj .
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860201.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860204.png" />, be a closed smooth Jordan curve in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860205.png" />-plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860206.png" /> be the complex density in a Cauchy-type integral along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860207.png" />, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s0860208.png" /> satisfies a Hölder condition:
+
Let $  \Gamma $:  
 +
$  t= t( s) $,  
 +
0\leq  s \leq  l $,  
 +
$  t( 0)= t( l) $,  
 +
be a closed smooth Jordan curve in the complex $  z $-
 +
plane, let $  \phi ( t) $
 +
be the complex density in a Cauchy-type integral along $  \Gamma $,  
 +
on which $  \phi ( t) $
 +
satisfies a Hölder condition:
 +
 
 +
$$
 +
| \phi ( t _ {1} ) - \phi ( t _ {2} ) |  \leq  C  | t _ {1} - t _ {2} |  ^  \alpha  ,\  0< \alpha \leq  1;
 +
$$
  
<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/s086/s086020/s0860209.png" /></td> </tr></table>
+
let  $  D  ^ {+} $
 +
be the interior of  $  \Gamma $,
 +
$  D  ^ {-} $
 +
its exterior, and let
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602010.png" /> be the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602012.png" /> its exterior, and let
+
$$ \tag{1 }
 +
\Phi ( z)  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( t)  dt }{t-}
 +
z ,\  z \notin \Gamma ,
 +
$$
  
<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/s086/s086020/s08602013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
be a Cauchy-type integral. Then, for any point  $  t _ {0} \in \Gamma $
 +
the limits
  
be a Cauchy-type integral. Then, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602014.png" /> the limits
+
$$
 +
\Phi  ^ {+} ( t _ {0} )  = \lim\limits _ {\begin{array}{c}
 +
z \rightarrow t _ {0} \\
  
<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/s086/s086020/s08602015.png" /></td> </tr></table>
+
z \in D  ^ {+}
 +
\end{array}
 +
}  \Phi ( 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/s086/s086020/s08602016.png" /></td> </tr></table>
+
$$
 +
\Phi  ^ {-} ( t _ {0} )  = \lim\limits _ {\begin{array}{c}
 +
z \rightarrow t _ {0} \\
 +
z \in D  ^ {-}
 +
\end{array}
 +
}  \Phi ( z)
 +
$$
  
 
exist and are given by the Sokhotskii formulas
 
exist and are given by the Sokhotskii 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/s086/s086020/s08602017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\left .
 +
\begin{array}{c}
 +
\Phi  ^ {+} ( t _ {0} )  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( t)  dt }{t- t _ {0} }
 +
+
 +
\frac{1}{2}
 +
\phi ( t _ {0} ),  \\
 +
\Phi  ^ {-} ( t _ {0} )  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( t)  dt }{t- t _ {0} }
 +
-  
 +
\frac{1}{2}
 +
\phi ( t _ {0} ),  \\
 +
\end{array}
 +
\right \}
 +
$$
  
 
or, equivalently,
 
or, equivalently,
  
<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/s086/s086020/s08602018.png" /></td> </tr></table>
+
$$
 +
\Phi  ^ {+} ( t _ {0} ) + \Phi  ^ {-} ( t _ {0} )  =
 +
\frac{1}{\pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( t)  dt }{t- t _ {0} }
 +
,
 +
$$
 +
 
 +
$$
 +
\Phi  ^ {+} ( t _ {0} ) - \Phi  ^ {-} ( t _ {0} )  = \phi ( t _ {0} ) .
 +
$$
 +
 
 +
The integrals along  $  \Gamma $
 +
on the right-hand sides of these formulas are understood in the sense of the Cauchy principal value and are so-called singular integrals. By taking, under these conditions,  $  \Phi  ^ {+} ( t) $(
 +
or  $  \Phi  ^ {-} ( t) $)
 +
as values of the integral  $  \Phi ( z) $
 +
on  $  \Gamma $,
 +
one thus obtains a function  $  \Phi ( z) $
 +
that is continuous in the closed domain  $  \overline{ {D  ^ {+} }}\; = D  ^ {+} \cup \Gamma $(
 +
respectively,  $  \overline{ {D  ^ {-} }}\; = D  ^ {-} \cup \Gamma $).  
 +
In the large,  $  \Phi ( z) $
 +
is sometimes described as a piecewise-analytic function.
 +
 
 +
If  $  \alpha < 1 $,
 +
then  $  \Phi  ^ {+} ( t) $
 +
and  $  \Phi  ^ {-} ( t) $
 +
are also Hölder continuous on  $  \Gamma $
 +
with the same exponent  $  \alpha $,
 +
while if  $  \alpha = 1 $,
 +
with any exponent  $  \alpha  ^  \prime  < 1 $.
 +
For the corner points  $  t _ {0} $
 +
of a piecewise-smooth curve  $  \Gamma $(
 +
see Fig.), the Sokhotskii formulas take 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/s086/s086020/s08602019.png" /></td> </tr></table>
+
$$ \tag{3 }
  
The integrals along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602020.png" /> on the right-hand sides of these formulas are understood in the sense of the Cauchy principal value and are so-called singular integrals. By taking, under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602021.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602022.png" />) as values of the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602024.png" />, one thus obtains a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602025.png" /> that is continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602026.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602027.png" />). In the large, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602028.png" /> is sometimes described as a piecewise-analytic function.
+
\begin{array}{c}
 +
\Phi  ^ {+} ( t _ {0} )  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( t)  dt }{t- t _ {0} }
 +
+ \left ( 1 -  
 +
\frac \beta {2 \pi }
 +
\right ) \phi ( t _ {0} ), \\
 +
\Phi  ^ {-} ( t _ {0} )  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( t) dt }{t- t _ {0} }
 +
-  
 +
\frac \beta {2 \pi }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602031.png" /> are also Hölder continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602032.png" /> with the same exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602033.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602034.png" />, with any exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602035.png" />. For the corner points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602036.png" /> of a piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602037.png" /> (see Fig.), the Sokhotskii formulas take the form
+
\phi ( t _ {0} ),0\leq  \beta \leq  2 \pi . \\
 +
\end{array}
  
<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/s086/s086020/s08602038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
  
In the case of a non-closed piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602039.png" />, the Sokhotskii formulas (2) and (3) remain valid for the interior points of the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602040.png" />.
+
In the case of a non-closed piecewise-smooth curve $  \Gamma $,  
 +
the Sokhotskii formulas (2) and (3) remain valid for the interior points of the arc $  \Gamma $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086020a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086020a.gif" />
Line 39: Line 150:
 
The Sokhotskii formulas play a basic role in solving boundary value problems of function theory and in the theory of singular integral equations (see [[#References|[3]]], [[#References|[5]]]), and also in solving various applied problems in function theory (see [[#References|[4]]]).
 
The Sokhotskii formulas play a basic role in solving boundary value problems of function theory and in the theory of singular integral equations (see [[#References|[3]]], [[#References|[5]]]), and also in solving various applied problems in function theory (see [[#References|[4]]]).
  
The question naturally arises as to the possibility of extending the conditions on the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602041.png" /> and the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602042.png" />, in such a way that the Sokhotskii formulas (possibly with some restrictions) remain valid. The most significant results in this direction are due to V.V. Golubev and I.I. Privalov (see [[#References|[6]]], [[#References|[8]]]). For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602043.png" /> be a rectifiable Jordan curve, and let the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602044.png" /> be, as before, Hölder continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602045.png" />. Then the Sokhotskii formulas (2) hold almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602048.png" /> are understood as non-tangential boundary values of the Cauchy-type integral from inside and outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602049.png" />, respectively, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602051.png" /> are, in general, not continuous in the closed domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602053.png" />.
+
The question naturally arises as to the possibility of extending the conditions on the contour $  \Gamma $
 +
and the density $  \phi ( t) $,  
 +
in such a way that the Sokhotskii formulas (possibly with some restrictions) remain valid. The most significant results in this direction are due to V.V. Golubev and I.I. Privalov (see [[#References|[6]]], [[#References|[8]]]). For example, let $  \Gamma $
 +
be a rectifiable Jordan curve, and let the density $  \phi ( t) $
 +
be, as before, Hölder continuous on $  \Gamma $.  
 +
Then the Sokhotskii formulas (2) hold almost everywhere on $  \Gamma $,  
 +
where $  \Phi  ^ {+} ( t _ {0} ) $
 +
and $  \Phi  ^ {-} ( t _ {0} ) $
 +
are understood as non-tangential boundary values of the Cauchy-type integral from inside and outside $  \Gamma $,  
 +
respectively, but $  \Phi  ^ {+} ( z) $
 +
and $  \Phi  ^ {-} ( z) $
 +
are, in general, not continuous in the closed domains $  \overline{ {D  ^ {+} }}\; $
 +
and $  \overline{ {D  ^ {-} }}\; $.
  
 
For spatial analogues of the Sokhotskii formulas see [[#References|[7]]].
 
For spatial analogues of the Sokhotskii formulas see [[#References|[7]]].
Line 45: Line 168:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Sokhotskii,  "On definite integrals and functions used in series expansions" , St. Petersburg  (1873)  (In Russian)  (Dissertation)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  J. Plemelj,  "Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend"  ''Monatsh. Math. Phys.'' , '''19'''  (1908)  pp. 205–210</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  J. Plemelj,  "Riemannsche Funktionenscharen mit gegebener Monodromiegruppe"  ''Monatsh. Math. Phys.'' , '''19'''  (1908)  pp. 211–245</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Some basic problems of the mathematical theory of elasticity" , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F.D. Gakhov,  "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</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">[8]</TD> <TD valign="top">  B.V. Khvedelidze,  "The method of Cauchy-type integrals in the discontinuous boundary value problems of the theory of holomorphic functions of a complex variable"  ''J. Soviet Math.'' , '''7'''  (1977)  pp. 309–415  ''Itogi Nauk. Sovremen. Probl. Mat.'' , '''7'''  (1975)  pp. 5–162</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Sokhotskii,  "On definite integrals and functions used in series expansions" , St. Petersburg  (1873)  (In Russian)  (Dissertation)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  J. Plemelj,  "Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend"  ''Monatsh. Math. Phys.'' , '''19'''  (1908)  pp. 205–210</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  J. Plemelj,  "Riemannsche Funktionenscharen mit gegebener Monodromiegruppe"  ''Monatsh. Math. Phys.'' , '''19'''  (1908)  pp. 211–245</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Some basic problems of the mathematical theory of elasticity" , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F.D. Gakhov,  "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</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">[8]</TD> <TD valign="top">  B.V. Khvedelidze,  "The method of Cauchy-type integrals in the discontinuous boundary value problems of the theory of holomorphic functions of a complex variable"  ''J. Soviet Math.'' , '''7'''  (1977)  pp. 309–415  ''Itogi Nauk. Sovremen. Probl. Mat.'' , '''7'''  (1975)  pp. 5–162</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
In the Western literature one usually speaks of the Plemelj formulas; the combination Sokhotskii–Plemelj formulas also occurs.
 
In the Western literature one usually speaks of the Plemelj formulas; the combination Sokhotskii–Plemelj formulas also occurs.
  
Much work has been done on Cauchy-type integrals in recent years. Thus, A.P. Calderón [[#References|[a1]]] has proved the existence of non-tangential boundary values almost everywhere under weaker conditions, and he has studied the principal value integral, or Cauchy transform, in (2) as a bounded linear operator on appropriate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602054.png" /> spaces. For further information see [[#References|[a2]]].
+
Much work has been done on Cauchy-type integrals in recent years. Thus, A.P. Calderón [[#References|[a1]]] has proved the existence of non-tangential boundary values almost everywhere under weaker conditions, and he has studied the principal value integral, or Cauchy transform, in (2) as a bounded linear operator on appropriate $  L _ {p} $
 +
spaces. For further information see [[#References|[a2]]].
  
The idea of representing a function on a simple closed curve as the jump of an analytic function across that curve has been extensively generalized. Thus, every distribution on the unit circle and every tempered distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086020/s08602055.png" /> can be represented as such jumps in an appropriate sense, cf. [[#References|[a3]]]. The ultimate in this direction are Sato's hyperfunctions (cf. [[#References|[a5]]] and [[Hyperfunction|Hyperfunction]]).
+
The idea of representing a function on a simple closed curve as the jump of an analytic function across that curve has been extensively generalized. Thus, every distribution on the unit circle and every tempered distribution on $  \mathbf R $
 +
can be represented as such jumps in an appropriate sense, cf. [[#References|[a3]]]. The ultimate in this direction are Sato's hyperfunctions (cf. [[#References|[a5]]] and [[Hyperfunction|Hyperfunction]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Calderón,  "Cauchy integrals on Lipschitz curves and related operators"  ''Proc. Nat. Acad. Sci. USA'' , '''74''' :  4  (1977)  pp. 1324–1327</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.P. Khavin (ed.)  S.V. Khrushchev (ed.)  N.K. Nikol'skii (ed.) , ''Linear and complex analysis problem book'' , ''Lect. notes in math.'' , '''1043''' , Springer  (1984)  pp. Chapt. 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.G. Tillmann,  "Darstellung der Schwartzschen Distributionen durch analytischer Funktionen"  ''Math. Z.'' , '''77'''  (1961)  pp. 106–124</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Sato,  "Theory of hyperfunctions I-II"  ''J. Fac. Sci. Univ. Tokyo I'' , '''8'''  (1959–1960)  pp. 139–193; 387–437</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Calderón,  "Cauchy integrals on Lipschitz curves and related operators"  ''Proc. Nat. Acad. Sci. USA'' , '''74''' :  4  (1977)  pp. 1324–1327</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.P. Khavin (ed.)  S.V. Khrushchev (ed.)  N.K. Nikol'skii (ed.) , ''Linear and complex analysis problem book'' , ''Lect. notes in math.'' , '''1043''' , Springer  (1984)  pp. Chapt. 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.G. Tillmann,  "Darstellung der Schwartzschen Distributionen durch analytischer Funktionen"  ''Math. Z.'' , '''77'''  (1961)  pp. 106–124</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Sato,  "Theory of hyperfunctions I-II"  ''J. Fac. Sci. Univ. Tokyo I'' , '''8'''  (1959–1960)  pp. 139–193; 387–437</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>

Latest revision as of 14:55, 7 June 2020


Formulas first discovered by Yu.V. Sokhotskii [1], expressing the boundary values of a Cauchy-type integral. With more complete proofs but significantly later, the formulas were obtained independently by J. Plemelj .

Let $ \Gamma $: $ t= t( s) $, $ 0\leq s \leq l $, $ t( 0)= t( l) $, be a closed smooth Jordan curve in the complex $ z $- plane, let $ \phi ( t) $ be the complex density in a Cauchy-type integral along $ \Gamma $, on which $ \phi ( t) $ satisfies a Hölder condition:

$$ | \phi ( t _ {1} ) - \phi ( t _ {2} ) | \leq C | t _ {1} - t _ {2} | ^ \alpha ,\ 0< \alpha \leq 1; $$

let $ D ^ {+} $ be the interior of $ \Gamma $, $ D ^ {-} $ its exterior, and let

$$ \tag{1 } \Phi ( z) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( t) dt }{t-} z ,\ z \notin \Gamma , $$

be a Cauchy-type integral. Then, for any point $ t _ {0} \in \Gamma $ the limits

$$ \Phi ^ {+} ( t _ {0} ) = \lim\limits _ {\begin{array}{c} z \rightarrow t _ {0} \\ z \in D ^ {+} \end{array} } \Phi ( z), $$

$$ \Phi ^ {-} ( t _ {0} ) = \lim\limits _ {\begin{array}{c} z \rightarrow t _ {0} \\ z \in D ^ {-} \end{array} } \Phi ( z) $$

exist and are given by the Sokhotskii formulas

$$ \tag{2 } \left . \begin{array}{c} \Phi ^ {+} ( t _ {0} ) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( t) dt }{t- t _ {0} } + \frac{1}{2} \phi ( t _ {0} ), \\ \Phi ^ {-} ( t _ {0} ) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( t) dt }{t- t _ {0} } - \frac{1}{2} \phi ( t _ {0} ), \\ \end{array} \right \} $$

or, equivalently,

$$ \Phi ^ {+} ( t _ {0} ) + \Phi ^ {-} ( t _ {0} ) = \frac{1}{\pi i } \int\limits _ \Gamma \frac{\phi ( t) dt }{t- t _ {0} } , $$

$$ \Phi ^ {+} ( t _ {0} ) - \Phi ^ {-} ( t _ {0} ) = \phi ( t _ {0} ) . $$

The integrals along $ \Gamma $ on the right-hand sides of these formulas are understood in the sense of the Cauchy principal value and are so-called singular integrals. By taking, under these conditions, $ \Phi ^ {+} ( t) $( or $ \Phi ^ {-} ( t) $) as values of the integral $ \Phi ( z) $ on $ \Gamma $, one thus obtains a function $ \Phi ( z) $ that is continuous in the closed domain $ \overline{ {D ^ {+} }}\; = D ^ {+} \cup \Gamma $( respectively, $ \overline{ {D ^ {-} }}\; = D ^ {-} \cup \Gamma $). In the large, $ \Phi ( z) $ is sometimes described as a piecewise-analytic function.

If $ \alpha < 1 $, then $ \Phi ^ {+} ( t) $ and $ \Phi ^ {-} ( t) $ are also Hölder continuous on $ \Gamma $ with the same exponent $ \alpha $, while if $ \alpha = 1 $, with any exponent $ \alpha ^ \prime < 1 $. For the corner points $ t _ {0} $ of a piecewise-smooth curve $ \Gamma $( see Fig.), the Sokhotskii formulas take the form

$$ \tag{3 } \begin{array}{c} \Phi ^ {+} ( t _ {0} ) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( t) dt }{t- t _ {0} } + \left ( 1 - \frac \beta {2 \pi } \right ) \phi ( t _ {0} ), \\ \Phi ^ {-} ( t _ {0} ) = \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( t) dt }{t- t _ {0} } - \frac \beta {2 \pi } \phi ( t _ {0} ),\ 0\leq \beta \leq 2 \pi . \\ \end{array} $$

In the case of a non-closed piecewise-smooth curve $ \Gamma $, the Sokhotskii formulas (2) and (3) remain valid for the interior points of the arc $ \Gamma $.

Figure: s086020a

The Sokhotskii formulas play a basic role in solving boundary value problems of function theory and in the theory of singular integral equations (see [3], [5]), and also in solving various applied problems in function theory (see [4]).

The question naturally arises as to the possibility of extending the conditions on the contour $ \Gamma $ and the density $ \phi ( t) $, in such a way that the Sokhotskii formulas (possibly with some restrictions) remain valid. The most significant results in this direction are due to V.V. Golubev and I.I. Privalov (see [6], [8]). For example, let $ \Gamma $ be a rectifiable Jordan curve, and let the density $ \phi ( t) $ be, as before, Hölder continuous on $ \Gamma $. Then the Sokhotskii formulas (2) hold almost everywhere on $ \Gamma $, where $ \Phi ^ {+} ( t _ {0} ) $ and $ \Phi ^ {-} ( t _ {0} ) $ are understood as non-tangential boundary values of the Cauchy-type integral from inside and outside $ \Gamma $, respectively, but $ \Phi ^ {+} ( z) $ and $ \Phi ^ {-} ( z) $ are, in general, not continuous in the closed domains $ \overline{ {D ^ {+} }}\; $ and $ \overline{ {D ^ {-} }}\; $.

For spatial analogues of the Sokhotskii formulas see [7].

References

[1] Yu.V. Sokhotskii, "On definite integrals and functions used in series expansions" , St. Petersburg (1873) (In Russian) (Dissertation)
[2a] J. Plemelj, "Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreffend" Monatsh. Math. Phys. , 19 (1908) pp. 205–210
[2b] J. Plemelj, "Riemannsche Funktionenscharen mit gegebener Monodromiegruppe" Monatsh. Math. Phys. , 19 (1908) pp. 211–245
[3] N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[4] N.I. Muskhelishvili, "Some basic problems of the mathematical theory of elasticity" , Noordhoff (1975) (Translated from Russian)
[5] F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)
[6] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[7] A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)
[8] B.V. Khvedelidze, "The method of Cauchy-type integrals in the discontinuous boundary value problems of the theory of holomorphic functions of a complex variable" J. Soviet Math. , 7 (1977) pp. 309–415 Itogi Nauk. Sovremen. Probl. Mat. , 7 (1975) pp. 5–162

Comments

In the Western literature one usually speaks of the Plemelj formulas; the combination Sokhotskii–Plemelj formulas also occurs.

Much work has been done on Cauchy-type integrals in recent years. Thus, A.P. Calderón [a1] has proved the existence of non-tangential boundary values almost everywhere under weaker conditions, and he has studied the principal value integral, or Cauchy transform, in (2) as a bounded linear operator on appropriate $ L _ {p} $ spaces. For further information see [a2].

The idea of representing a function on a simple closed curve as the jump of an analytic function across that curve has been extensively generalized. Thus, every distribution on the unit circle and every tempered distribution on $ \mathbf R $ can be represented as such jumps in an appropriate sense, cf. [a3]. The ultimate in this direction are Sato's hyperfunctions (cf. [a5] and Hyperfunction).

References

[a1] A.P. Calderón, "Cauchy integrals on Lipschitz curves and related operators" Proc. Nat. Acad. Sci. USA , 74 : 4 (1977) pp. 1324–1327
[a2] V.P. Khavin (ed.) S.V. Khrushchev (ed.) N.K. Nikol'skii (ed.) , Linear and complex analysis problem book , Lect. notes in math. , 1043 , Springer (1984) pp. Chapt. 6 (Translated from Russian)
[a3] H.G. Tillmann, "Darstellung der Schwartzschen Distributionen durch analytischer Funktionen" Math. Z. , 77 (1961) pp. 106–124
[a4] M. Sato, "Theory of hyperfunctions I-II" J. Fac. Sci. Univ. Tokyo I , 8 (1959–1960) pp. 139–193; 387–437
[a5] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)
How to Cite This Entry:
Sokhotskii formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sokhotskii_formulas&oldid=49591
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article