Namespaces
Variants
Actions

Difference between revisions of "Cauchy integral"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A Cauchy integral is a definite integral of a continuous function of one real variable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208901.png" /> be a continuous function on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208902.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208905.png" />. The limit
+
<!--
 +
c0208901.png
 +
$#A+1 = 247 n = 0
 +
$#C+1 = 247 : ~/encyclopedia/old_files/data/C020/C.0200890 Cauchy integral
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/c/c020/c020890/c0208906.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
is called the definite integral in Cauchy's sense of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208907.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c0208908.png" /> and is denoted by
+
A Cauchy integral is a definite integral of a continuous function of one real variable. Let  $  f (x) $
 +
be a continuous function on an interval  $  [a, b] $
 +
and let  $  a = x _ {0} < \dots < x _ {i - 1 }  < x _ {i} < \dots < x _ {n} = b $,
 +
$  \Delta x _ {i} = x _ {i} - x _ {i - 1 }  $,
 +
$  i = 1 \dots n $.  
 +
The limit
  
<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/c/c020/c020890/c0208909.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\max  \Delta x _ {i} \rightarrow 0 } \
 +
\sum _ {i = 1 } ^ { n }
 +
f (x _ {i - 1 }  )
 +
\Delta x _ {i}  $$
 +
 
 +
is called the definite integral in Cauchy's sense of  $  f (x) $
 +
over  $  [a, b] $
 +
and is denoted by
 +
 
 +
$$
 +
\int\limits _ { a } ^ { b }
 +
f (x)  dx.
 +
$$
  
 
The Cauchy integral is a particular case of the [[Riemann integral|Riemann integral]]. The definition is due to A.L. Cauchy [[#References|[1]]].
 
The Cauchy integral is a particular case of the [[Riemann integral|Riemann integral]]. The definition is due to A.L. Cauchy [[#References|[1]]].
Line 14: Line 40:
 
A Cauchy integral is an integral with the Cauchy kernel,
 
A Cauchy integral is an integral with the Cauchy kernel,
  
<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/c/c020/c020890/c02089010.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{1}{2 \pi i ( \zeta - z) }
 +
,
 +
$$
 +
 
 +
expressing the values of a regular analytic function  $  f (z) $
 +
in the interior of a contour  $  L $
 +
in terms of its values on  $  L $.
 +
More precisely: Let  $  f (z) $
 +
be a regular analytic function of the complex variable  $  z $
 +
in a domain  $  D $
 +
and let  $  L $
 +
be a closed piecewise-smooth Jordan curve lying in  $  D $
 +
together with its interior  $  G $;  
 +
it is assumed that  $  L $
 +
is described in the counter-clockwise sense. Then one has the following formula, which is of fundamental importance in the theory of analytic functions of one complex variable and which is known as the Cauchy integral formula:
 +
 
 +
$$ \tag{1 }
 +
f (z)  = \
 +
 
 +
\frac{1}{2 \pi i }
  
expressing the values of a regular analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089011.png" /> in the interior of a contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089012.png" /> in terms of its values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089013.png" />. More precisely: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089014.png" /> be a regular analytic function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089015.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089016.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089017.png" /> be a closed piecewise-smooth Jordan curve lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089018.png" /> together with its interior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089019.png" />; it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089020.png" /> is described in the counter-clockwise sense. Then one has the following formula, which is of fundamental importance in the theory of analytic functions of one complex variable and which is known as the Cauchy integral formula:
+
\int\limits _ { L }
  
<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/c/c020/c020890/c02089021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{f ( \zeta )  d \zeta }{\zeta - z }
 +
.
 +
$$
  
 
The integral on the right of (1) is also called a Cauchy integral.
 
The integral on the right of (1) is also called a Cauchy integral.
Line 24: Line 73:
 
Apparently, the Cauchy integral first appeared, in certain special cases, in the work of A.L. Cauchy [[#References|[1]]].
 
Apparently, the Cauchy integral first appeared, in certain special cases, in the work of A.L. Cauchy [[#References|[1]]].
  
Cauchy integrals are thus characterized by two conditions: 1) they are evaluated along a closed, smooth (or, at least, piecewise-smooth) curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089022.png" />; and 2) their integrands have the form
+
Cauchy integrals are thus characterized by two conditions: 1) they are evaluated along a closed, smooth (or, at least, piecewise-smooth) curve $  L $;  
 +
and 2) their integrands have the form
 +
 
 +
$$
 +
 
 +
\frac{f ( \zeta ) }{2 \pi i ( \zeta - z) }
 +
,
 +
$$
 +
 
 +
where  $  \zeta \in L $
 +
and  $  f (z) $
 +
is a regular analytic function on  $  L $
 +
and in the interior of  $  L $.
 +
If  $  z \in C \overline{G}\; $(
 +
the complement to  $  \overline{G}\; $)
 +
in the Cauchy integral, i.e. if  $  z $
 +
lies outside  $  L $,
 +
then, provided that the conditions 1) and 2) remain valid,
 +
 
 +
$$ \tag{2 }
  
<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/c/c020/c020890/c02089023.png" /></td> </tr></table>
+
\frac{1}{2 \pi i }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089025.png" /> is a regular analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089026.png" /> and in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089028.png" /> (the complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089029.png" />) in the Cauchy integral, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089030.png" /> lies outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089031.png" />, then, provided that the conditions 1) and 2) remain valid,
+
\int\limits _ { L }
  
<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/c/c020/c020890/c02089032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{f ( \zeta )  d \zeta }{\zeta - z }
 +
  = 0,\ \
 +
z \in C \overline{G}\; .
 +
$$
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089033.png" /> is the circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089034.png" /> centred at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089035.png" />, i.e.
+
In particular, if $  L $
 +
is the circle of radius $  \rho $
 +
centred at a point $  z $,  
 +
i.e.
  
<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/c/c020/c020890/c02089036.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ {\zeta = z + \rho e ^ {i \theta } } : {
 +
0 \leq  \theta < 2 \pi } \}
 +
,
 +
$$
  
 
then (1) implies that
 
then (1) implies that
  
<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/c/c020/c020890/c02089037.png" /></td> </tr></table>
+
$$
 +
f (z)  = \
 +
 
 +
\frac{1}{2 \pi i }
  
i.e. the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089038.png" /> at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089039.png" /> is equal to the arithmetic average of its values on any sufficiently small circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089040.png" /> centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089041.png" />. Formula (1) enables one to prove all other elementary properties of analytic functions.
+
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
f (z + \rho e ^ {i \theta } ) \
 +
d \theta ,
 +
$$
  
On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089042.png" /> is a regular analytic function in the infinite domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089043.png" /> (the exterior of the closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089044.png" />) and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089045.png" />, and if one defines
+
i.e. the value of  $  f (z) $
 +
at any point  $  z \in D $
 +
is equal to the arithmetic average of its values on any sufficiently small circle  $  L \subset  D $
 +
centred at  $  z $.  
 +
Formula (1) enables one to prove all other elementary properties of analytic functions.
  
<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/c/c020/c020890/c02089046.png" /></td> </tr></table>
+
On the other hand, if  $  f (z) $
 +
is a regular analytic function in the infinite domain  $  C \overline{G}\; $(
 +
the exterior of the closed curve  $  L $)
 +
and on  $  L $,
 +
and if one defines
 +
 
 +
$$
 +
f ( \infty )  = \
 +
\lim\limits _ {z \rightarrow \infty }
 +
f (z),
 +
$$
  
 
then the following formula, known as the Cauchy integral formula for an infinite domain, is valid:
 
then the following formula, known as the Cauchy integral formula for an infinite domain, is valid:
  
<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/c/c020/c020890/c02089047.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _ { L }
 +
 
 +
\frac{f ( \zeta )  d \zeta }{\zeta - z }
 +
  = \
 +
\left \{
 +
 
 +
\begin{array}{ll}
 +
f ( \infty ) - f (z),  &z \in C \overline{G}\; ,  \\
 +
f ( \infty ),  &z \in G. \\
 +
\end{array}
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089048.png" /> be some (not necessarily closed) piecewise-smooth curve in the finite plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089049.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089050.png" /> be a continuous complex function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089051.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089052.png" /> be a point not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089053.png" />. The term integral of Cauchy type is applied to the following generalization of the Cauchy integral:
+
\right .$$
  
<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/c/c020/c020890/c02089054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
Now let  $  \Gamma $
 +
be some (not necessarily closed) piecewise-smooth curve in the finite plane,  $  z \neq \infty $,
 +
let  $  \phi ( \zeta ) $
 +
be a continuous complex function on  $  \Gamma $
 +
and let  $  z $
 +
be a point not on  $  \Gamma $.  
 +
The term integral of Cauchy type is applied to the following generalization of the Cauchy integral:
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089055.png" /> is called the density of the integral of Cauchy type. Elementary properties of integrals of Cauchy type are:
+
$$ \tag{3 }
 +
F (z)  = \
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089056.png" /> is a regular analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089057.png" /> in any domain not containing points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089058.png" />.
+
\frac{1}{2 \pi i }
  
2) The derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089059.png" /> are given by the formulas
+
\int\limits _  \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/c/c020/c020890/c02089060.png" /></td> </tr></table>
+
\frac{\phi ( \zeta )  d \zeta }{\zeta - z }
 +
,\ \
 +
z \notin \Gamma .
 +
$$
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089061.png" /> is regular at infinity, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089063.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089064.png" />.
+
The function  $  \phi ( \zeta ) $
 +
is called the density of the integral of Cauchy type. Elementary properties of integrals of Cauchy type are:
  
From the point of view of the general theory of analytic functions and its applications to mechanics and physics, it is of fundamental importance to consider the existence of boundary values of an integral of Cauchy type as one approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089065.png" />, and to find analytic expressions for these values. The Cauchy integral (1) is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089066.png" /> everywhere in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089067.png" /> and vanishes identically outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089068.png" />. Therefore, when an integral of Cauchy type (3) reduces to a Cauchy integral, i.e. when the conditions 1) and 2) are satisfied, then, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089069.png" /> is approached from the left (i.e. from its interior), the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089070.png" /> has boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089071.png" />, and if these values are assumed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089072.png" /> it is continuous from the left on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089073.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089074.png" />; as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089075.png" /> is approached from the right (i.e. from its exterior), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089076.png" /> has boundary values zero, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089077.png" />, and if these values are assumed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089078.png" /> it is continuous from the right on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089079.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089080.png" />. Thus, for a Cauchy integral
+
1) $  F (z) $
 +
is a regular analytic function of  $  z $
 +
in any domain not containing points of  $  \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/c/c020/c020890/c02089081.png" /></td> </tr></table>
+
2) The derivatives  $  F ^ { (n) } (z) $
 +
are given by the formulas
  
For an integral of Cauchy type of general form the matter is somewhat more complicated. Suppose that the equation of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089082.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089083.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089084.png" /> denotes the arc length reckoned from some fixed point, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089085.png" /> be an arbitrary fixed point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089086.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089087.png" /> be the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089088.png" /> that remains after the smaller of the arcs with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089090.png" /> is deleted from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089091.png" />. If the limit
+
$$
 +
F ^ { (n) } (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/c/c020/c020890/c02089092.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{n! }{2 \pi i }
  
exists and is finite, it is called a singular integral. It can be proved, for example, that a singular integral (4) exists if the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089093.png" /> is smooth in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089094.png" /> distinct from the end points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089095.png" /> and if the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089096.png" /> satisfies a Hölder condition
+
\int\limits _  \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/c/c020/c020890/c02089097.png" /></td> </tr></table>
+
\frac{\phi ( \zeta )  d \zeta }{( \zeta - z) ^ {n + 1 } }
 +
,\ \
 +
z \notin \Gamma ; \ \
 +
n = 0, 1 , .  .  . .
 +
$$
 +
 
 +
3)  $  F (z) $
 +
is regular at infinity, with  $  F ( \infty ) = 0 $,
 +
$  F (z) = O (1/z) $
 +
as  $  z \rightarrow \infty $.
 +
 
 +
From the point of view of the general theory of analytic functions and its applications to mechanics and physics, it is of fundamental importance to consider the existence of boundary values of an integral of Cauchy type as one approaches  $  \Gamma $,
 +
and to find analytic expressions for these values. The Cauchy integral (1) is equal to  $  f (z) $
 +
everywhere in the interior of  $  L $
 +
and vanishes identically outside  $  L $.
 +
Therefore, when an integral of Cauchy type (3) reduces to a Cauchy integral, i.e. when the conditions 1) and 2) are satisfied, then, as  $  L $
 +
is approached from the left (i.e. from its interior), the function  $  F (z) $
 +
has boundary values  $  F  ^ {+} ( \zeta _ {0} ) = f ( \zeta _ {0} ) $,
 +
and if these values are assumed on  $  L $
 +
it is continuous from the left on  $  L $
 +
at each point  $  \zeta _ {0} \in L $;  
 +
as  $  L $
 +
is approached from the right (i.e. from its exterior), then  $  F (z) $
 +
has boundary values zero, i.e.  $  F  ^ {-} ( \zeta _ {0} ) = 0 $,
 +
and if these values are assumed on  $  L $
 +
it is continuous from the right on  $  L $
 +
at each point  $  \zeta _ {0} \in L $.
 +
Thus, for a Cauchy integral
 +
 
 +
$$
 +
F  ^ {+} ( \zeta _ {0} ) -
 +
F  ^ {-} ( \zeta _ {0} )  = \
 +
f ( \zeta _ {0} ).
 +
$$
 +
 
 +
For an integral of Cauchy type of general form the matter is somewhat more complicated. Suppose that the equation of the curve  $  \Gamma $
 +
is  $  \zeta = \zeta (s) $,
 +
where  $  s $
 +
denotes the arc length reckoned from some fixed point, let  $  \zeta _ {0} = \zeta (s _ {0} ) $
 +
be an arbitrary fixed point on  $  \Gamma $
 +
and let  $  \Gamma _  \epsilon  $
 +
be the part of  $  \Gamma $
 +
that remains after the smaller of the arcs with end points  $  \zeta (s _ {0} - \epsilon ) $
 +
and  $  \zeta (s _ {0} + \epsilon ) $
 +
is deleted from  $  \Gamma $.  
 +
If the limit
 +
 
 +
$$ \tag{4 }
 +
\lim\limits _ {\epsilon \rightarrow 0 }
 +
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _ {\Gamma _  \epsilon  }
 +
 
 +
\frac{\phi ( \zeta )  d \zeta }{\zeta - \zeta _ {0} }
 +
  = \
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _  \Gamma
 +
 
 +
\frac{\phi ( \zeta )  d \zeta }{\zeta - \zeta _ {0} }
 +
,\ \
 +
\zeta _ {0} \in \Gamma ,
 +
$$
 +
 
 +
exists and is finite, it is called a singular integral. It can be proved, for example, that a singular integral (4) exists if the curve  $  \Gamma $
 +
is smooth in a neighbourhood of a point  $  \zeta _ {0} $
 +
distinct from the end points of  $  \Gamma $
 +
and if the density  $  \phi ( \zeta ) $
 +
satisfies a Hölder condition
 +
 
 +
$$
 +
| \phi ( \zeta  ^  \prime  ) -
 +
\phi ( \zeta  ^ {\prime\prime} ) |  \leq  \
 +
C  | \zeta  ^  \prime  - \zeta  ^ {\prime\prime} |  ^  \mu  ,\ \
 +
\mu > 0.
 +
$$
  
 
Under these conditions there also exist boundary values, and these are given by the Sokhotskii formulas:
 
Under these conditions there also exist boundary values, and these 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/c/c020/c020890/c02089098.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
F ^ { \pm  } ( \zeta _ {0} )  = \
 +
\pm  {
 +
\frac{1}{2}
 +
}
 +
\phi ( \zeta _ {0} ) +
  
and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c02089099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890100.png" /> are continuous in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890101.png" /> from the left and right, respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890102.png" />. In the case of a Cauchy integral, the singular integral is equal to
+
\frac{1}{2 \pi i }
  
<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/c/c020/c020890/c020890103.png" /></td> </tr></table>
+
\int\limits _  \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/c/c020/c020890/c020890104.png" /></td> </tr></table>
+
\frac{\phi ( \zeta )  d \zeta }{\zeta - \zeta _ {0} }
 +
,\ \
 +
\zeta _ {0} \in \Gamma ,
 +
$$
 +
 
 +
and the functions  $  F  ^ {+} (z) $
 +
and  $  F  ^ {-} (z) $
 +
are continuous in a neighbourhood of  $  \zeta _ {0} \in \Gamma $
 +
from the left and right, respectively, of  $  \Gamma $.
 +
In the case of a Cauchy integral, the singular integral is equal to
 +
 
 +
$$
 +
 
 +
\frac{f ( \zeta _ {0} ) }{2}
 +
,
 +
$$
 +
 
 +
$$
 +
F  ^ {+} ( \zeta _ {0} ) - F  ^ {-} ( \zeta _ {0} )  = \
 +
f ( \zeta _ {0} ) ,\  F  ^ {+} ( \zeta _ {0} )
 +
+ F  ^ {-} ( \zeta _ {0} )  = f ( \zeta _ {0} ).
 +
$$
  
 
An equivalent form of (5) is
 
An equivalent form of (5) is
  
<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/c/c020/c020890/c020890105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
F  ^ {+} ( \zeta _ {0} ) -
 +
F  ^ {-} ( \zeta _ {0} )  = \
 +
\phi ( \zeta _ {0} ),
 +
$$
  
<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/c/c020/c020890/c020890106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
F  ^ {+} ( \zeta _ {0} ) + F  ^ {-} ( \zeta _ {0} )
 +
=
 +
\frac{1}{\pi i }
 +
\int\limits _  \Gamma 
 +
\frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} }
 +
,\  \zeta _ {0} \in \Gamma .
 +
$$
  
 
The Sokhotskii formulas (5)–(7) are of fundamental importance in the solution of [[Boundary value problems of analytic function theory|boundary value problems of analytic function theory]], of singular integral equations connected with integrals of Cauchy type (cf. [[Singular integral equation|Singular integral equation]]), and also in the solution of various problems in hydrodynamics, elasticity theory, etc.
 
The Sokhotskii formulas (5)–(7) are of fundamental importance in the solution of [[Boundary value problems of analytic function theory|boundary value problems of analytic function theory]], of singular integral equations connected with integrals of Cauchy type (cf. [[Singular integral equation|Singular integral equation]]), and also in the solution of various problems in hydrodynamics, elasticity theory, etc.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890107.png" /> be an arbitrary rectifiable curve of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890108.png" />; for simplicity it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890109.png" /> is closed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890110.png" /> be the angle between the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890111.png" />-axis and the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890112.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890113.png" />, regarded as a function of the arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890114.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890115.png" /> be a complex function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890116.png" /> of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890117.png" />. The expression
+
Let $  \Gamma $
 +
be an arbitrary rectifiable curve of length $  l $;  
 +
for simplicity it is assumed that $  \Gamma $
 +
is closed. Let $  \psi = \psi (s) $
 +
be the angle between the direction of the $  x $-
 +
axis and the tangent to $  \Gamma $
 +
at the point $  \zeta = \zeta (s) \in \Gamma $,  
 +
regarded as a function of the arc length $  s $,  
 +
and let $  \Phi (s) $
 +
be a complex function of $  s $
 +
of bounded variation on $  [0, l] $.  
 +
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/c/c020/c020890/c020890118.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
F (z) = \
  
is called an integral of Cauchy–Stieltjes type. In other words, an integral of Cauchy–Stieltjes type is an integral of Cauchy type with respect to an arbitrary finite complex Borel measure with support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890119.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890120.png" /> is absolutely continuous, then the integral of Cauchy–Stieltjes type becomes an integral of Cauchy–Lebesgue type, often called simply an integral of Cauchy type:
+
\frac{1}{2 \pi i }
  
<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/c/c020/c020890/c020890121.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
\int\limits _  \Gamma
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890122.png" />.
+
\frac{e ^ {i \psi }  d \Phi (s) }{\zeta - z }
 +
,\ \
 +
\zeta = \zeta (s),\ \
 +
z \notin \Gamma ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890123.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890124.png" /> at which there exists a well-defined tangent, inclined to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890125.png" />-axis at an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890126.png" />; such points exist almost-everywhere on a rectifiable curve. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890127.png" /> be the point on the straight line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890128.png" /> and inclined to the normal at an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890129.png" />, at a distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890130.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890131.png" />. The difference between the integral of Cauchy–Stieltjes type (8) and the integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890132.png" />,
+
is called an integral of Cauchy–Stieltjes type. In other words, an integral of Cauchy–Stieltjes type is an integral of Cauchy type with respect to an arbitrary finite complex Borel measure with support on $  \Gamma $.  
 +
If  $  \Phi (s) $
 +
is absolutely continuous, then the integral of Cauchy–Stieltjes type becomes an integral of Cauchy–Lebesgue type, often called simply an integral of Cauchy type:
  
<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/c/c020/c020890/c020890133.png" /></td> </tr></table>
+
$$ \tag{9 }
 +
F (z)  = \
  
is defined at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890134.png" /> where the tangent is defined, i.e. almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890135.png" />. An important proposition in the theory of integrals of Cauchy–Stieltjes type is Privalov's fundamental lemma: The limit
+
\frac{1}{2 \pi i }
  
<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/c/c020/c020890/c020890136.png" /></td> </tr></table>
+
\int\limits _  \Gamma
  
exists for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890137.png" />, with the possible exception of a point set of measure zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890138.png" />, independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890139.png" />; the convergence is uniform in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890140.png" /> in any angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890142.png" />. If the singular integral exists almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890143.png" />, then the integral of Cauchy–Stieltjes type has angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890144.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890145.png" /> and these satisfy the Sokhotskii formulas:
+
\frac{\phi ( \zeta )  d \zeta }{\zeta - 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/c/c020/c020890/c020890146.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>
+
where  $  \phi ( \zeta ) = \phi [ \zeta (s)] = \Phi  ^  \prime  (s) $.
  
The converse is also true: If an integral of Cauchy–Stieltjes type has angular boundary values from both inside and outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890147.png" />, almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890148.png" />, then the singular integral exists and formulas (10) are valid almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890149.png" />. As yet (1987) there is no complete solution to the problem of finding reasonably simple necessary and sufficient conditions for the existence of boundary values for integrals of Cauchy–Stieltjes type or even for integrals of Cauchy–Lebesgue type.
+
Let  $  \zeta _ {0} $
 +
be a point of  $  \Gamma $
 +
at which there exists a well-defined tangent, inclined to the  $  x $-
 +
axis at an angle  $  \psi _ {0} $;
 +
such points exist almost-everywhere on a rectifiable curve. Let  $  z $
 +
be the point on the straight line passing through  $  \zeta _ {0} $
 +
and inclined to the normal at an angle  $  \alpha _ {0} $,
 +
at a distance  $  | z - \zeta _ {0} | = \epsilon $,
 +
i.e. $  z = \zeta _ {0} \pm  \epsilon i e ^ {i ( \psi _ {0} + \alpha _ {0} ) } $.
 +
The difference between the integral of Cauchy–Stieltjes type (8) and the integral over  $  \Gamma _  \epsilon  $,
  
In contrast to the previously considered case of an integral of Cauchy type over a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890150.png" />, an integral of Cauchy–Stieltjes type, even when it has angular boundary values, is no longer necessarily a continuous function in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890151.png" /> from the left or right of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890152.png" />. It is known, for example, that an integral of Cauchy–Lebesgue type (9) is continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890153.png" /> bounded by the rectifiable contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890154.png" />, provided one additionally assumes that the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890155.png" /> satisfies a Lipschitz condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890156.png" />:
+
$$
 +
W ( \zeta _ {0} ;  \epsilon , \alpha _ {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/c/c020/c020890/c020890157.png" /></td> </tr></table>
+
\frac{1}{2 \pi i }
 +
 
 +
\left [ \int\limits _  \Gamma
 +
 
 +
\frac{e ^ {i \psi }  d \Phi (s) }{\zeta - z }
 +
-
 +
\int\limits _ {\Gamma _  \epsilon  }
 +
 
 +
\frac{e ^ {i \psi }  d \Phi (s) }{\zeta - \zeta _ {0} }
 +
 
 +
\right ] ,
 +
$$
 +
 
 +
is defined at all points  $  \zeta _ {0} \in \Gamma $
 +
where the tangent is defined, i.e. almost-everywhere on  $  \Gamma $.
 +
An important proposition in the theory of integrals of Cauchy–Stieltjes type is Privalov's fundamental lemma: The limit
 +
 
 +
$$
 +
\lim\limits _ {\epsilon \rightarrow 0 } \
 +
W ( \zeta _ {0} ;  \epsilon , \alpha _ {0} )  = \
 +
\pm  {
 +
\frac{1}{2}
 +
} \Phi  ^  \prime  ( \zeta _ {0} )
 +
$$
 +
 
 +
exists for all points  $  \zeta _ {0} \in \Gamma $,
 +
with the possible exception of a point set of measure zero on  $  \Gamma $,
 +
independent of  $  \alpha _ {0} $;  
 +
the convergence is uniform in  $  \alpha _ {0} $
 +
in any angle  $  | \alpha _ {0} | < \pi /2 - \delta $,
 +
$  \delta > 0 $.
 +
If the singular integral exists almost-everywhere on  $  \Gamma $,
 +
then the integral of Cauchy–Stieltjes type has angular boundary values  $  F ^ { \pm  } ( \zeta _ {0} ) $
 +
almost-everywhere on  $  \Gamma $
 +
and these satisfy the Sokhotskii formulas:
 +
 
 +
$$ \tag{10 }
 +
F  ^  \pm  ( \zeta _ {0} )  = \
 +
\pm  {
 +
\frac{1}{2}
 +
}
 +
\Phi  ^  \prime  (s _ {0} ) +
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _  \Gamma
 +
 
 +
\frac{e ^ {i \psi }  d \Phi (s) }{\zeta - \zeta _ {0} }
 +
,\ \
 +
\zeta _ {0} \in \Gamma .
 +
$$
 +
 
 +
The converse is also true: If an integral of Cauchy–Stieltjes type has angular boundary values from both inside and outside  $  \Gamma $,
 +
almost-everywhere on  $  \Gamma $,
 +
then the singular integral exists and formulas (10) are valid almost-everywhere on  $  \Gamma $.  
 +
As yet (1987) there is no complete solution to the problem of finding reasonably simple necessary and sufficient conditions for the existence of boundary values for integrals of Cauchy–Stieltjes type or even for integrals of Cauchy–Lebesgue type.
 +
 
 +
In contrast to the previously considered case of an integral of Cauchy type over a smooth curve  $  \Gamma $,
 +
an integral of Cauchy–Stieltjes type, even when it has angular boundary values, is no longer necessarily a continuous function in a neighbourhood of  $  \zeta _ {0} \in \Gamma $
 +
from the left or right of  $  \Gamma $.
 +
It is known, for example, that an integral of Cauchy–Lebesgue type (9) is continuous in the closed domain  $  \overline{D}\; $
 +
bounded by the rectifiable contour  $  \Gamma $,
 +
provided one additionally assumes that the density  $  \phi ( \zeta ) $
 +
satisfies a Lipschitz condition on  $  \Gamma $:
 +
 
 +
$$
 +
| \phi ( \zeta  ^  \prime  ) -
 +
\phi ( \zeta  ^ {\prime\prime} ) |
 +
\leq  C  | \zeta  ^  \prime  -
 +
\zeta  ^ {\prime\prime} |,\ \
 +
\zeta  ^  \prime  ,\
 +
\zeta  ^ {\prime\prime} \in \Gamma .
 +
$$
  
 
One says that an integral of Cauchy–Lebesgue type (9) becomes a Cauchy integral
 
One says that an integral of Cauchy–Lebesgue type (9) becomes a Cauchy integral
  
<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/c/c020/c020890/c020890158.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>
+
$$ \tag{11 }
 +
F (z) = \
  
in the sense of Lebesgue, if the angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890159.png" /> from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890160.png" /> coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890161.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890162.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890163.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890164.png" />. In this context the Golubev–Privalov theorem holds: A summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890165.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890166.png" /> represents the angular boundary values of some Cauchy integral from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890167.png" /> if and only if all its moments vanish:
+
\frac{1}{2 \pi i }
  
<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/c/c020/c020890/c020890168.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>
+
\int\limits _  \Gamma
 +
 
 +
\frac{\phi ( \zeta )  d \zeta }{\zeta - z }
 +
,
 +
$$
 +
 
 +
in the sense of Lebesgue, if the angular boundary values  $  F  ^ {+} ( \zeta _ {0} ) $
 +
from the inside of  $  \Gamma $
 +
coincide with  $  \phi ( \zeta _ {0} ) $
 +
almost-everywhere on  $  \Gamma $,
 +
i.e. $  F  ^ {-} ( \zeta _ {0} ) = 0 $
 +
almost-everywhere on  $  \Gamma $.
 +
In this context the Golubev–Privalov theorem holds: A summable function  $  \phi ( \zeta ) $
 +
on  $  \Gamma $
 +
represents the angular boundary values of some Cauchy integral from the inside of  $  \Gamma $
 +
if and only if all its moments vanish:
 +
 
 +
$$ \tag{12 }
 +
\int\limits _  \Gamma
 +
\zeta  ^ {n} \phi ( \zeta ) \
 +
d \zeta  =  0,\ \
 +
n = 0, 1 , .  .  . .
 +
$$
  
 
If the analogous conditions
 
If the analogous conditions
  
<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/c/c020/c020890/c020890169.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>
+
$$ \tag{13 }
 +
\int\limits _  \Gamma
 +
\zeta  ^ {n} e ^ {i \psi }
 +
d \Phi (s)  = 0,\ \
 +
n = 0, 1 \dots
 +
$$
  
 
are satisfied, then the integral of Cauchy–Stieltjes type (8) becomes a Cauchy–Stieltjes integral:
 
are satisfied, then the integral of Cauchy–Stieltjes type (8) becomes a Cauchy–Stieltjes integral:
  
<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/c/c020/c020890/c020890170.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>
+
$$ \tag{14 }
 +
F (z)  = \
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _  \Gamma
 +
 
 +
\frac{e ^ {i \psi }  d \Phi (s) }{\zeta - z }
 +
,
 +
$$
 +
 
 +
i.e. the angular boundary values  $  F  ^ {+} ( \zeta _ {0} ) $
 +
from the inside of  $  \Gamma $
 +
coincide with the derivative  $  \Phi  ^  \prime  (s _ {0} ) $
 +
almost-everywhere on  $  \Gamma $,
 +
or, stated differently, the angular boundary values  $  F  ^ {-} ( \zeta _ {0} ) $
 +
from the outside of  $  \Gamma $
 +
vanish almost-everywhere on  $  \Gamma $.
 +
Conditions (13) immediately imply that the function  $  \Phi (s) $
 +
is absolutely continuous on  $  [0, l] $
 +
and, consequently, in this case the Cauchy–Stieltjes integral (14) is in fact a Cauchy–Lebesgue integral with density  $  \phi ( \zeta ) = \phi [ \zeta (s)] = \Phi  ^  \prime  (s) $.
 +
Thus, the class of functions representable by a Cauchy–Stieltjes integral is identical with the class of functions representable by a Cauchy–Lebesgue integral.
 +
 
 +
An important problem is the intrinsic characterization of classes of functions which are regular in a domain  $  D $
 +
bounded by a closed rectifiable curve  $  \Gamma $,
 +
and representable by a Cauchy integral (11), an integral of Cauchy–Lebesgue type (9), or an integral of Cauchy–Stieltjes type (8); concerning the most important classes  $  A (D) $,
 +
$  B (D) = H _  \infty  (D) $,
 +
$  H _ {p} (D) $
 +
and  $  N  ^ {*} (D) $
 +
see [[Boundary properties of analytic functions|Boundary properties of analytic functions]].
 +
 
 +
In the simplest case, when  $  D = \{ {z } : {| z | < 1 } \} $
 +
is the unit disc and  $  \Gamma = \{ {z } : {| z | = 1 } \} $
 +
is the unit circle, an integral of Cauchy–Stieltjes type, which in this case has the form
 +
 
 +
$$ \tag{15 }
 +
F (z)  = \
 +
 
 +
\frac{1}{2 \pi }
 +
 
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
 
 +
\frac{\zeta  d \Phi ( \theta ) }{\zeta - z }
 +
,\ \
 +
| z | < 1,\ \
 +
\zeta = e ^ {i \theta } ,
 +
$$
  
i.e. the angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890171.png" /> from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890172.png" /> coincide with the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890173.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890174.png" />, or, stated differently, the angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890175.png" /> from the outside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890176.png" /> vanish almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890177.png" />. Conditions (13) immediately imply that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890178.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890179.png" /> and, consequently, in this case the Cauchy–Stieltjes integral (14) is in fact a Cauchy–Lebesgue integral with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890180.png" />. Thus, the class of functions representable by a Cauchy–Stieltjes integral is identical with the class of functions representable by a Cauchy–Lebesgue integral.
+
always represents a function of class  $  H _ {p} $,
 +
0 < p < 1 $.  
 +
The converse is false: The set of functions of classes  $  H _ {p} $,
 +
0 < p < 1 $,  
 +
is more extensive than the set of functions representable in the form (15). On the other hand, the set of functions representable in  $  D $
 +
by a Cauchy–Stieltjes or a Cauchy integral is identical with the class $  H _ {1} $.
  
An important problem is the intrinsic characterization of classes of functions which are regular in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890181.png" /> bounded by a closed rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890182.png" />, and representable by a Cauchy integral (11), an integral of Cauchy–Lebesgue type (9), or an integral of Cauchy–Stieltjes type (8); concerning the most important classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890183.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890185.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890186.png" /> see [[Boundary properties of analytic functions|Boundary properties of analytic functions]].
+
In the case of an arbitrary simply-connected domain $  D $
 +
bounded by a rectifiable curve $  \Gamma $,  
 +
the class of functions representable in  $  D $
 +
by a Cauchy–Stieltjes or a Cauchy integral is identical with the Smirnov class  $  E _ {1} $(
 +
see [[Boundary properties of analytic functions|Boundary properties of analytic functions]]). The characteristics of the classes of functions representable by an integral of Cauchy–Stieltjes type or an integral of Cauchy–Lebesgue type are considerably more complicated.
  
In the simplest case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890187.png" /> is the unit disc and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890188.png" /> is the unit circle, an integral of Cauchy–Stieltjes type, which in this case has the form
+
Let  $  f (z) $
 +
be an arbitrary (non-analytic) function of class  $  C  ^ {1} $
 +
in a finite closed domain  $  \overline{D}\; $
 +
bounded by a piecewise-smooth Jordan curve  $  L $.  
 +
The term Cauchy integral formula is sometimes applied also to the following generalization of the classical formula (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/c/c020/c020890/c020890189.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>
+
$$ \tag{16 }
  
always represents a function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890191.png" />. The converse is false: The set of functions of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890193.png" />, is more extensive than the set of functions representable in the form (15). On the other hand, the set of functions representable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890194.png" /> by a Cauchy–Stieltjes or a Cauchy integral is identical with the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890195.png" />.
+
\frac{1}{2 \pi i }
  
In the case of an arbitrary simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890196.png" /> bounded by a rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890197.png" />, the class of functions representable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890198.png" /> by a Cauchy–Stieltjes or a Cauchy integral is identical with the Smirnov class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890199.png" /> (see [[Boundary properties of analytic functions|Boundary properties of analytic functions]]). The characteristics of the classes of functions representable by an integral of Cauchy–Stieltjes type or an integral of Cauchy–Lebesgue type are considerably more complicated.
+
\int\limits _ { L }
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890200.png" /> be an arbitrary (non-analytic) function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890201.png" /> in a finite closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890202.png" /> bounded by a piecewise-smooth Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890203.png" />. The term Cauchy integral formula is sometimes applied also to the following generalization of the classical formula (1):
+
\frac{f ( \zeta )  d \zeta }{\zeta - z }
 +
-
 +
{
 +
\frac{1} \pi
 +
}
 +
{\int\limits \int\limits } _ { D }
  
<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/c/c020/c020890/c020890204.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>
+
\frac{\partial  f }{\partial  \overline \zeta \; }
  
<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/c/c020/c020890/c020890205.png" /></td> </tr></table>
+
\frac{d \xi  d \eta }{\zeta - z }
 +
=
 +
$$
 +
 
 +
$$
 +
= \
 +
\left \{
 +
\begin{array}{ll}
 +
f (z),  & z \in D,  \\
 +
0,  & z \in C \overline{D}\; ,  \\
 +
\end{array}
 +
 
 +
\right .$$
  
 
where
 
where
  
<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/c/c020/c020890/c020890206.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  f }{\partial  \overline \zeta \; }
 +
  = \
 +
{
 +
\frac{1}{2}
 +
} \left (
 +
 
 +
\frac{\partial  f }{\partial  \xi }
 +
+ i
 +
 
 +
\frac{\partial  f }{\partial  \eta }
 +
 
 +
\right ) ,\ \
 +
\zeta = \xi + i \eta .
 +
$$
  
 
This formula first appeared, apparently, in the work of D. Pompeiu (1912). It is also known as the Pompeiu formula, the Borel–Pompeiu formula, or the Cauchy–Green formula, and is widely applied in the theory of generalized analytic functions, singular integral equations and various applied problems.
 
This formula first appeared, apparently, in the work of D. Pompeiu (1912). It is also known as the Pompeiu formula, the Borel–Pompeiu formula, or the Cauchy–Green formula, and is widely applied in the theory of generalized analytic functions, singular integral equations and various applied problems.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890207.png" /> be a regular analytic function of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890208.png" /> in a closed polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890210.png" />. Then, at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890211.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890212.png" /> is representable by a multiple Cauchy integral:
+
Let $  f (z) $
 +
be a regular analytic function of several complex variables $  z = (z _ {1} \dots z _ {n} ) $
 +
in a closed polydisc $  \overline{D}\; $,
 +
$  D = \{ {z \in \mathbf C  ^ {n} } : {| z _  \nu  - a _  \nu  | < r _  \nu  } \} $.  
 +
Then, at each point of $  D $,  
 +
$  f (z) $
 +
is representable by a multiple Cauchy integral:
  
<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/c/c020/c020890/c020890213.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>
+
$$ \tag{17 }
 +
f (z) = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890214.png" /> is the distinguished boundary of the polydisc, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890215.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890216.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890217.png" />. Formula (17) yields a simple analogue of the Cauchy integral for a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890218.png" />, but when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890219.png" /> the integration in (17) extends not over the entire boundary of the polydisc but only over its distinguished boundary. In general, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890220.png" /> be a polycircular domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890221.png" /> — a product of simply-connected plane domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890222.png" /> with smooth boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890223.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890224.png" /> be the distinguished boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890225.png" />, which is a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890226.png" />-dimensional manifold. Formula (17) also generalizes to this case.
+
\frac{1}{(2 \pi i) ^ {n} }
  
More profound generalizations of the Cauchy integral formula are extremely important in the theory of analytic functions of several complex variables; such generalizations are the [[Leray formula|Leray formula]] (which J. Leray himself called the Cauchy–Fantappié formula) and the [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]]. In this connection, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890227.png" /> the theory is concerned mainly with boundary properties of integral representations other than (17).
+
\int\limits _ { T }
 +
 
 +
\frac{f ( \zeta )  d \zeta }{\zeta - z }
 +
,
 +
$$
 +
 
 +
where  $  T = \{ {\zeta \in \mathbf C  ^ {n} } : {| \zeta _  \nu  - a _  \nu  | = r _  \nu  ,  \nu = 1 \dots n } \} $
 +
is the distinguished boundary of the polydisc,  $  \zeta = ( \zeta _ {1} \dots \zeta _ {n} ) $,
 +
$  d \zeta = d \zeta _ {1} \dots d \zeta _ {n} $,
 +
$  \zeta - z = ( \zeta _ {1} - z _ {1} ) \dots ( \zeta _ {n} - z _ {n} ) $.
 +
Formula (17) yields a simple analogue of the Cauchy integral for a circle  $  L = \{ {z \in \mathbf C } : {| z - a | = r } \} $,
 +
but when  $  n > 1 $
 +
the integration in (17) extends not over the entire boundary of the polydisc but only over its distinguished boundary. In general, let  $  D = D _ {1} \times \dots \times D _ {n} $
 +
be a polycircular domain in  $  \mathbf C  ^ {n} $—
 +
a product of simply-connected plane domains  $  D _  \nu  $
 +
with smooth boundaries  $  \partial  D _  \nu  = \{ {z _  \nu  = z _  \nu  ( t _  \nu  ) } : {0 \leq  t _  \nu  \leq  1 } \} $;
 +
let  $  T = \partial  D _ {1} \times \dots \times \partial  D _ {n} $
 +
be the distinguished boundary of  $  D $,
 +
which is a smooth  $  n $-
 +
dimensional manifold. Formula (17) also generalizes to this case.
 +
 
 +
More profound generalizations of the Cauchy integral formula are extremely important in the theory of analytic functions of several complex variables; such generalizations are the [[Leray formula|Leray formula]] (which J. Leray himself called the Cauchy–Fantappié formula) and the [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]]. In this connection, when $  n > 1 $
 +
the theory is concerned mainly with boundary properties of integral representations other than (17).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Sur la mécanique céleste et sur un nouveau calcul appelé calcul des limites" , Turin  (1831)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1–3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.I. Privalov,  "The Cauchy integral" , Saratov  (1918)  (In Russian)</TD></TR><TR><TD valign="top">[7]</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">[8]</TD> <TD valign="top">  S.Ya. Khavinson,  ''Itogi Nauk. Mat. Anal. 1963''  (1965)  pp. 5–80</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.V. Khvedelidze,  "The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of a complex variable" , ''Contemporary problems in mathematics'' , '''7''' , Moscow  (1975)  pp. 5–162  (In Russian)</TD></TR><TR><TD valign="top">[10]</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Sur la mécanique céleste et sur un nouveau calcul appelé calcul des limites" , Turin  (1831)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1–3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.I. Privalov,  "The Cauchy integral" , Saratov  (1918)  (In Russian)</TD></TR><TR><TD valign="top">[7]</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">[8]</TD> <TD valign="top">  S.Ya. Khavinson,  ''Itogi Nauk. Mat. Anal. 1963''  (1965)  pp. 5–80</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.V. Khvedelidze,  "The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of a complex variable" , ''Contemporary problems in mathematics'' , '''7''' , Moscow  (1975)  pp. 5–162  (In Russian)</TD></TR><TR><TD valign="top">[10]</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
In the non-Soviet literature Plemelj formulas is the usual name for what is here called Sokhotskii formulas.
 
In the non-Soviet literature Plemelj formulas is the usual name for what is here called Sokhotskii formulas.
  
Mapping properties of the singular integral operator associated to integrals of Cauchy type form an important subject. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890228.png" /> be the graph of a Lipschitz function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890229.png" />. The principal result, due to A.P. Calderón and in full generality to G. David, is that the singular integral operator
+
Mapping properties of the singular integral operator associated to integrals of Cauchy type form an important subject. Let $  \Gamma $
 +
be the graph of a Lipschitz function $  \phi (x) $.  
 +
The principal result, due to A.P. Calderón and in full generality to G. David, is that the singular integral operator
  
<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/c/c020/c020890/c020890230.png" /></td> </tr></table>
+
$$
 +
f  \rightarrow \
 +
\int\limits _  \Gamma
  
at first defined as a principal value integral for compactly supported smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890231.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890232.png" />, extends to a bounded linear operator sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890233.png" /> to itself, and (hence) also sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890234.png" /> to itself <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890235.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890236.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890237.png" />, the functions of bounded mean oscillation.
+
\frac{f ( \zeta ) }{z - \zeta }
 +
\
 +
d \zeta ,
 +
$$
 +
 
 +
at first defined as a principal value integral for compactly supported smooth functions $  f $
 +
on $  \Gamma $,  
 +
extends to a bounded linear operator sending $  L _ {2} ( \Gamma ) $
 +
to itself, and (hence) also sending $  L _ {p} ( \Gamma ) $
 +
to itself $  (1 < p < \infty ) $
 +
and $  L _  \infty  ( \Gamma ) $
 +
to $  \mathop{\rm BMO} $,  
 +
the functions of bounded mean oscillation.
  
 
Formally one can write:
 
Formally one can write:
  
<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/c/c020/c020890/c020890238.png" /></td> </tr></table>
+
$$
 +
\int\limits _  \Gamma
 +
 
 +
\frac{f ( \zeta ) }{z - \zeta }
 +
\
 +
d \zeta  = \
 +
\int\limits _ {\mathbf R }
 +
 
 +
\frac{f ( \xi + i \phi ( \xi )) }{x - \xi + i ( \phi (x) - \phi ( \xi )) }
 +
 
 +
(1 + \phi  ^  \prime  ( \xi )) \
 +
d \xi =
 +
$$
 +
 
 +
$$
 +
= \
 +
\sum _ { 0 } ^  \infty  \int\limits _ {\mathbf R }
 +
\frac{f ( \xi + i
 +
\phi ( \xi )) (1 + i \phi ( \xi )) }{x - \xi }
 +
 
 +
\left (
 +
\frac{(-i) ( \phi (x) - \phi ( \xi )) }{x - \xi }
 +
\right )  ^ {n}  d \xi .
 +
$$
 +
 
 +
The integral operators  $  C _ {n} ( \phi ) $
 +
with kernel
  
<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/c/c020/c020890/c020890239.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{1}{x - \xi }
 +
}
 +
\left (
  
The integral operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890240.png" /> with kernel
+
\frac{\phi (x) - \phi ( \xi ) }{x - \xi }
  
<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/c/c020/c020890/c020890241.png" /></td> </tr></table>
+
\right )  ^ {n}
 +
$$
  
are the so-called commutators of Calderón. These are of independent interest, e.g. in the theory of partial differential equations (cf. [[Differential equation, partial|Differential equation, partial]]). The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890242.png" /> have the same mapping properties as the Cauchy integral operator, as was shown by R.R. Coifman, A. McIntosh and Y. Meyer. The best norm estimate known at this moment (1987) is that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890243.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890244.png" /> such that
+
are the so-called commutators of Calderón. These are of independent interest, e.g. in the theory of partial differential equations (cf. [[Differential equation, partial|Differential equation, partial]]). The operators $  C _ {n} ( \phi ) $
 +
have the same mapping properties as the Cauchy integral operator, as was shown by R.R. Coifman, A. McIntosh and Y. Meyer. The best norm estimate known at this moment (1987) is that for every $  \delta > 0 $
 +
there exists a c _  \delta  > 0 $
 +
such that
  
<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/c/c020/c020890/c020890245.png" /></td> </tr></table>
+
$$
 +
\| C _ {n} ( \phi ) \|  \leq  \
 +
c _  \delta  (n + 1) ^ {1 + \delta } \
 +
\| \phi  ^  \prime  \| _  \infty  ^ {n} .
 +
$$
  
 
This estimate was obtained by M. Christ and J.L. Journé.
 
This estimate was obtained by M. Christ and J.L. Journé.
Line 202: Line 747:
 
The Cauchy integral operators as well as Calderón's commutators are examples of so-called Calderón–Zygmund operators. For these results and further references see [[#References|[10]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]].
 
The Cauchy integral operators as well as Calderón's commutators are examples of so-called Calderón–Zygmund operators. For these results and further references see [[#References|[10]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]].
  
For results concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890246.png" /> functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020890/c020890247.png" />, which can be represented by Cauchy integrals, see [[#References|[a1]]].
+
For results concerning $  H _ {p} $
 +
functions, $  0 < p < 1 $,  
 +
which can be represented by Cauchy integrals, see [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.B. Aleksandrov,  "Essays on non locally convex Hardy classes"  V.P. Havin [V.P. Khavin] (ed.)  N.K. Nikol'skii (ed.) , ''Complex analysis and spectral theory'' , Springer  (1981)  pp. 1–89</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Christ,  J.L. Journé,  "Estimates for multilinear singular integral operators with polynomial growth"  (1986)  (Preprint Dept. of Math. Princeton Univ.)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.R. Coifman,  Y. Meyer,  "Non linear harmonic analysis, operator theory and P.D.E."  E.M. Stein (ed.) , ''Beijing lectures in harmonic analysis'' , Princeton Univ. Press  (1986)  pp. 3–46</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Journé,  "Calderón–Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.B. Aleksandrov,  "Essays on non locally convex Hardy classes"  V.P. Havin [V.P. Khavin] (ed.)  N.K. Nikol'skii (ed.) , ''Complex analysis and spectral theory'' , Springer  (1981)  pp. 1–89</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Christ,  J.L. Journé,  "Estimates for multilinear singular integral operators with polynomial growth"  (1986)  (Preprint Dept. of Math. Princeton Univ.)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.R. Coifman,  Y. Meyer,  "Non linear harmonic analysis, operator theory and P.D.E."  E.M. Stein (ed.) , ''Beijing lectures in harmonic analysis'' , Princeton Univ. Press  (1986)  pp. 3–46</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Journé,  "Calderón–Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón" , Springer  (1983)</TD></TR></table>

Revision as of 15:35, 4 June 2020


A Cauchy integral is a definite integral of a continuous function of one real variable. Let $ f (x) $ be a continuous function on an interval $ [a, b] $ and let $ a = x _ {0} < \dots < x _ {i - 1 } < x _ {i} < \dots < x _ {n} = b $, $ \Delta x _ {i} = x _ {i} - x _ {i - 1 } $, $ i = 1 \dots n $. The limit

$$ \lim\limits _ {\max \Delta x _ {i} \rightarrow 0 } \ \sum _ {i = 1 } ^ { n } f (x _ {i - 1 } ) \Delta x _ {i} $$

is called the definite integral in Cauchy's sense of $ f (x) $ over $ [a, b] $ and is denoted by

$$ \int\limits _ { a } ^ { b } f (x) dx. $$

The Cauchy integral is a particular case of the Riemann integral. The definition is due to A.L. Cauchy [1].

References

[1] A.L. Cauchy, "Résumé des leçons données à l'Ecole Royale Polytechnique sur le calcul infinitésimal" , 1 , Paris (1823)

A Cauchy integral is an integral with the Cauchy kernel,

$$ \frac{1}{2 \pi i ( \zeta - z) } , $$

expressing the values of a regular analytic function $ f (z) $ in the interior of a contour $ L $ in terms of its values on $ L $. More precisely: Let $ f (z) $ be a regular analytic function of the complex variable $ z $ in a domain $ D $ and let $ L $ be a closed piecewise-smooth Jordan curve lying in $ D $ together with its interior $ G $; it is assumed that $ L $ is described in the counter-clockwise sense. Then one has the following formula, which is of fundamental importance in the theory of analytic functions of one complex variable and which is known as the Cauchy integral formula:

$$ \tag{1 } f (z) = \ \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } . $$

The integral on the right of (1) is also called a Cauchy integral.

Apparently, the Cauchy integral first appeared, in certain special cases, in the work of A.L. Cauchy [1].

Cauchy integrals are thus characterized by two conditions: 1) they are evaluated along a closed, smooth (or, at least, piecewise-smooth) curve $ L $; and 2) their integrands have the form

$$ \frac{f ( \zeta ) }{2 \pi i ( \zeta - z) } , $$

where $ \zeta \in L $ and $ f (z) $ is a regular analytic function on $ L $ and in the interior of $ L $. If $ z \in C \overline{G}\; $( the complement to $ \overline{G}\; $) in the Cauchy integral, i.e. if $ z $ lies outside $ L $, then, provided that the conditions 1) and 2) remain valid,

$$ \tag{2 } \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } = 0,\ \ z \in C \overline{G}\; . $$

In particular, if $ L $ is the circle of radius $ \rho $ centred at a point $ z $, i.e.

$$ L = \ \{ {\zeta = z + \rho e ^ {i \theta } } : { 0 \leq \theta < 2 \pi } \} , $$

then (1) implies that

$$ f (z) = \ \frac{1}{2 \pi i } \int\limits _ { 0 } ^ { {2 } \pi } f (z + \rho e ^ {i \theta } ) \ d \theta , $$

i.e. the value of $ f (z) $ at any point $ z \in D $ is equal to the arithmetic average of its values on any sufficiently small circle $ L \subset D $ centred at $ z $. Formula (1) enables one to prove all other elementary properties of analytic functions.

On the other hand, if $ f (z) $ is a regular analytic function in the infinite domain $ C \overline{G}\; $( the exterior of the closed curve $ L $) and on $ L $, and if one defines

$$ f ( \infty ) = \ \lim\limits _ {z \rightarrow \infty } f (z), $$

then the following formula, known as the Cauchy integral formula for an infinite domain, is valid:

$$ \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } = \ \left \{ \begin{array}{ll} f ( \infty ) - f (z), &z \in C \overline{G}\; , \\ f ( \infty ), &z \in G. \\ \end{array} \right .$$

Now let $ \Gamma $ be some (not necessarily closed) piecewise-smooth curve in the finite plane, $ z \neq \infty $, let $ \phi ( \zeta ) $ be a continuous complex function on $ \Gamma $ and let $ z $ be a point not on $ \Gamma $. The term integral of Cauchy type is applied to the following generalization of the Cauchy integral:

$$ \tag{3 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - z } ,\ \ z \notin \Gamma . $$

The function $ \phi ( \zeta ) $ is called the density of the integral of Cauchy type. Elementary properties of integrals of Cauchy type are:

1) $ F (z) $ is a regular analytic function of $ z $ in any domain not containing points of $ \Gamma $.

2) The derivatives $ F ^ { (n) } (z) $ are given by the formulas

$$ F ^ { (n) } (z) = \ \frac{n! }{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{( \zeta - z) ^ {n + 1 } } ,\ \ z \notin \Gamma ; \ \ n = 0, 1 , . . . . $$

3) $ F (z) $ is regular at infinity, with $ F ( \infty ) = 0 $, $ F (z) = O (1/z) $ as $ z \rightarrow \infty $.

From the point of view of the general theory of analytic functions and its applications to mechanics and physics, it is of fundamental importance to consider the existence of boundary values of an integral of Cauchy type as one approaches $ \Gamma $, and to find analytic expressions for these values. The Cauchy integral (1) is equal to $ f (z) $ everywhere in the interior of $ L $ and vanishes identically outside $ L $. Therefore, when an integral of Cauchy type (3) reduces to a Cauchy integral, i.e. when the conditions 1) and 2) are satisfied, then, as $ L $ is approached from the left (i.e. from its interior), the function $ F (z) $ has boundary values $ F ^ {+} ( \zeta _ {0} ) = f ( \zeta _ {0} ) $, and if these values are assumed on $ L $ it is continuous from the left on $ L $ at each point $ \zeta _ {0} \in L $; as $ L $ is approached from the right (i.e. from its exterior), then $ F (z) $ has boundary values zero, i.e. $ F ^ {-} ( \zeta _ {0} ) = 0 $, and if these values are assumed on $ L $ it is continuous from the right on $ L $ at each point $ \zeta _ {0} \in L $. Thus, for a Cauchy integral

$$ F ^ {+} ( \zeta _ {0} ) - F ^ {-} ( \zeta _ {0} ) = \ f ( \zeta _ {0} ). $$

For an integral of Cauchy type of general form the matter is somewhat more complicated. Suppose that the equation of the curve $ \Gamma $ is $ \zeta = \zeta (s) $, where $ s $ denotes the arc length reckoned from some fixed point, let $ \zeta _ {0} = \zeta (s _ {0} ) $ be an arbitrary fixed point on $ \Gamma $ and let $ \Gamma _ \epsilon $ be the part of $ \Gamma $ that remains after the smaller of the arcs with end points $ \zeta (s _ {0} - \epsilon ) $ and $ \zeta (s _ {0} + \epsilon ) $ is deleted from $ \Gamma $. If the limit

$$ \tag{4 } \lim\limits _ {\epsilon \rightarrow 0 } \frac{1}{2 \pi i } \int\limits _ {\Gamma _ \epsilon } \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } ,\ \ \zeta _ {0} \in \Gamma , $$

exists and is finite, it is called a singular integral. It can be proved, for example, that a singular integral (4) exists if the curve $ \Gamma $ is smooth in a neighbourhood of a point $ \zeta _ {0} $ distinct from the end points of $ \Gamma $ and if the density $ \phi ( \zeta ) $ satisfies a Hölder condition

$$ | \phi ( \zeta ^ \prime ) - \phi ( \zeta ^ {\prime\prime} ) | \leq \ C | \zeta ^ \prime - \zeta ^ {\prime\prime} | ^ \mu ,\ \ \mu > 0. $$

Under these conditions there also exist boundary values, and these are given by the Sokhotskii formulas:

$$ \tag{5 } F ^ { \pm } ( \zeta _ {0} ) = \ \pm { \frac{1}{2} } \phi ( \zeta _ {0} ) + \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } ,\ \ \zeta _ {0} \in \Gamma , $$

and the functions $ F ^ {+} (z) $ and $ F ^ {-} (z) $ are continuous in a neighbourhood of $ \zeta _ {0} \in \Gamma $ from the left and right, respectively, of $ \Gamma $. In the case of a Cauchy integral, the singular integral is equal to

$$ \frac{f ( \zeta _ {0} ) }{2} , $$

$$ F ^ {+} ( \zeta _ {0} ) - F ^ {-} ( \zeta _ {0} ) = \ f ( \zeta _ {0} ) ,\ F ^ {+} ( \zeta _ {0} ) + F ^ {-} ( \zeta _ {0} ) = f ( \zeta _ {0} ). $$

An equivalent form of (5) is

$$ \tag{6 } F ^ {+} ( \zeta _ {0} ) - F ^ {-} ( \zeta _ {0} ) = \ \phi ( \zeta _ {0} ), $$

$$ \tag{7 } F ^ {+} ( \zeta _ {0} ) + F ^ {-} ( \zeta _ {0} ) = \frac{1}{\pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - \zeta _ {0} } ,\ \zeta _ {0} \in \Gamma . $$

The Sokhotskii formulas (5)–(7) are of fundamental importance in the solution of boundary value problems of analytic function theory, of singular integral equations connected with integrals of Cauchy type (cf. Singular integral equation), and also in the solution of various problems in hydrodynamics, elasticity theory, etc.

Let $ \Gamma $ be an arbitrary rectifiable curve of length $ l $; for simplicity it is assumed that $ \Gamma $ is closed. Let $ \psi = \psi (s) $ be the angle between the direction of the $ x $- axis and the tangent to $ \Gamma $ at the point $ \zeta = \zeta (s) \in \Gamma $, regarded as a function of the arc length $ s $, and let $ \Phi (s) $ be a complex function of $ s $ of bounded variation on $ [0, l] $. The expression

$$ \tag{8 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - z } ,\ \ \zeta = \zeta (s),\ \ z \notin \Gamma , $$

is called an integral of Cauchy–Stieltjes type. In other words, an integral of Cauchy–Stieltjes type is an integral of Cauchy type with respect to an arbitrary finite complex Borel measure with support on $ \Gamma $. If $ \Phi (s) $ is absolutely continuous, then the integral of Cauchy–Stieltjes type becomes an integral of Cauchy–Lebesgue type, often called simply an integral of Cauchy type:

$$ \tag{9 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - z } ,\ \ z \notin \Gamma , $$

where $ \phi ( \zeta ) = \phi [ \zeta (s)] = \Phi ^ \prime (s) $.

Let $ \zeta _ {0} $ be a point of $ \Gamma $ at which there exists a well-defined tangent, inclined to the $ x $- axis at an angle $ \psi _ {0} $; such points exist almost-everywhere on a rectifiable curve. Let $ z $ be the point on the straight line passing through $ \zeta _ {0} $ and inclined to the normal at an angle $ \alpha _ {0} $, at a distance $ | z - \zeta _ {0} | = \epsilon $, i.e. $ z = \zeta _ {0} \pm \epsilon i e ^ {i ( \psi _ {0} + \alpha _ {0} ) } $. The difference between the integral of Cauchy–Stieltjes type (8) and the integral over $ \Gamma _ \epsilon $,

$$ W ( \zeta _ {0} ; \epsilon , \alpha _ {0} ) = \ \frac{1}{2 \pi i } \left [ \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - z } - \int\limits _ {\Gamma _ \epsilon } \frac{e ^ {i \psi } d \Phi (s) }{\zeta - \zeta _ {0} } \right ] , $$

is defined at all points $ \zeta _ {0} \in \Gamma $ where the tangent is defined, i.e. almost-everywhere on $ \Gamma $. An important proposition in the theory of integrals of Cauchy–Stieltjes type is Privalov's fundamental lemma: The limit

$$ \lim\limits _ {\epsilon \rightarrow 0 } \ W ( \zeta _ {0} ; \epsilon , \alpha _ {0} ) = \ \pm { \frac{1}{2} } \Phi ^ \prime ( \zeta _ {0} ) $$

exists for all points $ \zeta _ {0} \in \Gamma $, with the possible exception of a point set of measure zero on $ \Gamma $, independent of $ \alpha _ {0} $; the convergence is uniform in $ \alpha _ {0} $ in any angle $ | \alpha _ {0} | < \pi /2 - \delta $, $ \delta > 0 $. If the singular integral exists almost-everywhere on $ \Gamma $, then the integral of Cauchy–Stieltjes type has angular boundary values $ F ^ { \pm } ( \zeta _ {0} ) $ almost-everywhere on $ \Gamma $ and these satisfy the Sokhotskii formulas:

$$ \tag{10 } F ^ \pm ( \zeta _ {0} ) = \ \pm { \frac{1}{2} } \Phi ^ \prime (s _ {0} ) + \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - \zeta _ {0} } ,\ \ \zeta _ {0} \in \Gamma . $$

The converse is also true: If an integral of Cauchy–Stieltjes type has angular boundary values from both inside and outside $ \Gamma $, almost-everywhere on $ \Gamma $, then the singular integral exists and formulas (10) are valid almost-everywhere on $ \Gamma $. As yet (1987) there is no complete solution to the problem of finding reasonably simple necessary and sufficient conditions for the existence of boundary values for integrals of Cauchy–Stieltjes type or even for integrals of Cauchy–Lebesgue type.

In contrast to the previously considered case of an integral of Cauchy type over a smooth curve $ \Gamma $, an integral of Cauchy–Stieltjes type, even when it has angular boundary values, is no longer necessarily a continuous function in a neighbourhood of $ \zeta _ {0} \in \Gamma $ from the left or right of $ \Gamma $. It is known, for example, that an integral of Cauchy–Lebesgue type (9) is continuous in the closed domain $ \overline{D}\; $ bounded by the rectifiable contour $ \Gamma $, provided one additionally assumes that the density $ \phi ( \zeta ) $ satisfies a Lipschitz condition on $ \Gamma $:

$$ | \phi ( \zeta ^ \prime ) - \phi ( \zeta ^ {\prime\prime} ) | \leq C | \zeta ^ \prime - \zeta ^ {\prime\prime} |,\ \ \zeta ^ \prime ,\ \zeta ^ {\prime\prime} \in \Gamma . $$

One says that an integral of Cauchy–Lebesgue type (9) becomes a Cauchy integral

$$ \tag{11 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{\phi ( \zeta ) d \zeta }{\zeta - z } , $$

in the sense of Lebesgue, if the angular boundary values $ F ^ {+} ( \zeta _ {0} ) $ from the inside of $ \Gamma $ coincide with $ \phi ( \zeta _ {0} ) $ almost-everywhere on $ \Gamma $, i.e. $ F ^ {-} ( \zeta _ {0} ) = 0 $ almost-everywhere on $ \Gamma $. In this context the Golubev–Privalov theorem holds: A summable function $ \phi ( \zeta ) $ on $ \Gamma $ represents the angular boundary values of some Cauchy integral from the inside of $ \Gamma $ if and only if all its moments vanish:

$$ \tag{12 } \int\limits _ \Gamma \zeta ^ {n} \phi ( \zeta ) \ d \zeta = 0,\ \ n = 0, 1 , . . . . $$

If the analogous conditions

$$ \tag{13 } \int\limits _ \Gamma \zeta ^ {n} e ^ {i \psi } d \Phi (s) = 0,\ \ n = 0, 1 \dots $$

are satisfied, then the integral of Cauchy–Stieltjes type (8) becomes a Cauchy–Stieltjes integral:

$$ \tag{14 } F (z) = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{e ^ {i \psi } d \Phi (s) }{\zeta - z } , $$

i.e. the angular boundary values $ F ^ {+} ( \zeta _ {0} ) $ from the inside of $ \Gamma $ coincide with the derivative $ \Phi ^ \prime (s _ {0} ) $ almost-everywhere on $ \Gamma $, or, stated differently, the angular boundary values $ F ^ {-} ( \zeta _ {0} ) $ from the outside of $ \Gamma $ vanish almost-everywhere on $ \Gamma $. Conditions (13) immediately imply that the function $ \Phi (s) $ is absolutely continuous on $ [0, l] $ and, consequently, in this case the Cauchy–Stieltjes integral (14) is in fact a Cauchy–Lebesgue integral with density $ \phi ( \zeta ) = \phi [ \zeta (s)] = \Phi ^ \prime (s) $. Thus, the class of functions representable by a Cauchy–Stieltjes integral is identical with the class of functions representable by a Cauchy–Lebesgue integral.

An important problem is the intrinsic characterization of classes of functions which are regular in a domain $ D $ bounded by a closed rectifiable curve $ \Gamma $, and representable by a Cauchy integral (11), an integral of Cauchy–Lebesgue type (9), or an integral of Cauchy–Stieltjes type (8); concerning the most important classes $ A (D) $, $ B (D) = H _ \infty (D) $, $ H _ {p} (D) $ and $ N ^ {*} (D) $ see Boundary properties of analytic functions.

In the simplest case, when $ D = \{ {z } : {| z | < 1 } \} $ is the unit disc and $ \Gamma = \{ {z } : {| z | = 1 } \} $ is the unit circle, an integral of Cauchy–Stieltjes type, which in this case has the form

$$ \tag{15 } F (z) = \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \frac{\zeta d \Phi ( \theta ) }{\zeta - z } ,\ \ | z | < 1,\ \ \zeta = e ^ {i \theta } , $$

always represents a function of class $ H _ {p} $, $ 0 < p < 1 $. The converse is false: The set of functions of classes $ H _ {p} $, $ 0 < p < 1 $, is more extensive than the set of functions representable in the form (15). On the other hand, the set of functions representable in $ D $ by a Cauchy–Stieltjes or a Cauchy integral is identical with the class $ H _ {1} $.

In the case of an arbitrary simply-connected domain $ D $ bounded by a rectifiable curve $ \Gamma $, the class of functions representable in $ D $ by a Cauchy–Stieltjes or a Cauchy integral is identical with the Smirnov class $ E _ {1} $( see Boundary properties of analytic functions). The characteristics of the classes of functions representable by an integral of Cauchy–Stieltjes type or an integral of Cauchy–Lebesgue type are considerably more complicated.

Let $ f (z) $ be an arbitrary (non-analytic) function of class $ C ^ {1} $ in a finite closed domain $ \overline{D}\; $ bounded by a piecewise-smooth Jordan curve $ L $. The term Cauchy integral formula is sometimes applied also to the following generalization of the classical formula (1):

$$ \tag{16 } \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } - { \frac{1} \pi } {\int\limits \int\limits } _ { D } \frac{\partial f }{\partial \overline \zeta \; } \frac{d \xi d \eta }{\zeta - z } = $$

$$ = \ \left \{ \begin{array}{ll} f (z), & z \in D, \\ 0, & z \in C \overline{D}\; , \\ \end{array} \right .$$

where

$$ \frac{\partial f }{\partial \overline \zeta \; } = \ { \frac{1}{2} } \left ( \frac{\partial f }{\partial \xi } + i \frac{\partial f }{\partial \eta } \right ) ,\ \ \zeta = \xi + i \eta . $$

This formula first appeared, apparently, in the work of D. Pompeiu (1912). It is also known as the Pompeiu formula, the Borel–Pompeiu formula, or the Cauchy–Green formula, and is widely applied in the theory of generalized analytic functions, singular integral equations and various applied problems.

Let $ f (z) $ be a regular analytic function of several complex variables $ z = (z _ {1} \dots z _ {n} ) $ in a closed polydisc $ \overline{D}\; $, $ D = \{ {z \in \mathbf C ^ {n} } : {| z _ \nu - a _ \nu | < r _ \nu } \} $. Then, at each point of $ D $, $ f (z) $ is representable by a multiple Cauchy integral:

$$ \tag{17 } f (z) = \ \frac{1}{(2 \pi i) ^ {n} } \int\limits _ { T } \frac{f ( \zeta ) d \zeta }{\zeta - z } , $$

where $ T = \{ {\zeta \in \mathbf C ^ {n} } : {| \zeta _ \nu - a _ \nu | = r _ \nu , \nu = 1 \dots n } \} $ is the distinguished boundary of the polydisc, $ \zeta = ( \zeta _ {1} \dots \zeta _ {n} ) $, $ d \zeta = d \zeta _ {1} \dots d \zeta _ {n} $, $ \zeta - z = ( \zeta _ {1} - z _ {1} ) \dots ( \zeta _ {n} - z _ {n} ) $. Formula (17) yields a simple analogue of the Cauchy integral for a circle $ L = \{ {z \in \mathbf C } : {| z - a | = r } \} $, but when $ n > 1 $ the integration in (17) extends not over the entire boundary of the polydisc but only over its distinguished boundary. In general, let $ D = D _ {1} \times \dots \times D _ {n} $ be a polycircular domain in $ \mathbf C ^ {n} $— a product of simply-connected plane domains $ D _ \nu $ with smooth boundaries $ \partial D _ \nu = \{ {z _ \nu = z _ \nu ( t _ \nu ) } : {0 \leq t _ \nu \leq 1 } \} $; let $ T = \partial D _ {1} \times \dots \times \partial D _ {n} $ be the distinguished boundary of $ D $, which is a smooth $ n $- dimensional manifold. Formula (17) also generalizes to this case.

More profound generalizations of the Cauchy integral formula are extremely important in the theory of analytic functions of several complex variables; such generalizations are the Leray formula (which J. Leray himself called the Cauchy–Fantappié formula) and the Bochner–Martinelli representation formula. In this connection, when $ n > 1 $ the theory is concerned mainly with boundary properties of integral representations other than (17).

References

[1] A.L. Cauchy, "Sur la mécanique céleste et sur un nouveau calcul appelé calcul des limites" , Turin (1831)
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[3] A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian)
[4] N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[5] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[6] I.I. Privalov, "The Cauchy integral" , Saratov (1918) (In Russian)
[7] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[8] S.Ya. Khavinson, Itogi Nauk. Mat. Anal. 1963 (1965) pp. 5–80
[9] B.V. Khvedelidze, "The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of a complex variable" , Contemporary problems in mathematics , 7 , Moscow (1975) pp. 5–162 (In Russian)
[10] A.P. Calderón, "Cauchy integrals on Lipschitz curves and related operators" Proc. Nat. Acad. Sci. USA , 74 : 4 (1977) pp. 1324–1327

Comments

In the non-Soviet literature Plemelj formulas is the usual name for what is here called Sokhotskii formulas.

Mapping properties of the singular integral operator associated to integrals of Cauchy type form an important subject. Let $ \Gamma $ be the graph of a Lipschitz function $ \phi (x) $. The principal result, due to A.P. Calderón and in full generality to G. David, is that the singular integral operator

$$ f \rightarrow \ \int\limits _ \Gamma \frac{f ( \zeta ) }{z - \zeta } \ d \zeta , $$

at first defined as a principal value integral for compactly supported smooth functions $ f $ on $ \Gamma $, extends to a bounded linear operator sending $ L _ {2} ( \Gamma ) $ to itself, and (hence) also sending $ L _ {p} ( \Gamma ) $ to itself $ (1 < p < \infty ) $ and $ L _ \infty ( \Gamma ) $ to $ \mathop{\rm BMO} $, the functions of bounded mean oscillation.

Formally one can write:

$$ \int\limits _ \Gamma \frac{f ( \zeta ) }{z - \zeta } \ d \zeta = \ \int\limits _ {\mathbf R } \frac{f ( \xi + i \phi ( \xi )) }{x - \xi + i ( \phi (x) - \phi ( \xi )) } (1 + \phi ^ \prime ( \xi )) \ d \xi = $$

$$ = \ \sum _ { 0 } ^ \infty \int\limits _ {\mathbf R } \frac{f ( \xi + i \phi ( \xi )) (1 + i \phi ( \xi )) }{x - \xi } \left ( \frac{(-i) ( \phi (x) - \phi ( \xi )) }{x - \xi } \right ) ^ {n} d \xi . $$

The integral operators $ C _ {n} ( \phi ) $ with kernel

$$ { \frac{1}{x - \xi } } \left ( \frac{\phi (x) - \phi ( \xi ) }{x - \xi } \right ) ^ {n} $$

are the so-called commutators of Calderón. These are of independent interest, e.g. in the theory of partial differential equations (cf. Differential equation, partial). The operators $ C _ {n} ( \phi ) $ have the same mapping properties as the Cauchy integral operator, as was shown by R.R. Coifman, A. McIntosh and Y. Meyer. The best norm estimate known at this moment (1987) is that for every $ \delta > 0 $ there exists a $ c _ \delta > 0 $ such that

$$ \| C _ {n} ( \phi ) \| \leq \ c _ \delta (n + 1) ^ {1 + \delta } \ \| \phi ^ \prime \| _ \infty ^ {n} . $$

This estimate was obtained by M. Christ and J.L. Journé.

The Cauchy integral operators as well as Calderón's commutators are examples of so-called Calderón–Zygmund operators. For these results and further references see [10], [a2], [a3], [a4].

For results concerning $ H _ {p} $ functions, $ 0 < p < 1 $, which can be represented by Cauchy integrals, see [a1].

References

[a1] A.B. Aleksandrov, "Essays on non locally convex Hardy classes" V.P. Havin [V.P. Khavin] (ed.) N.K. Nikol'skii (ed.) , Complex analysis and spectral theory , Springer (1981) pp. 1–89
[a2] M. Christ, J.L. Journé, "Estimates for multilinear singular integral operators with polynomial growth" (1986) (Preprint Dept. of Math. Princeton Univ.)
[a3] R.R. Coifman, Y. Meyer, "Non linear harmonic analysis, operator theory and P.D.E." E.M. Stein (ed.) , Beijing lectures in harmonic analysis , Princeton Univ. Press (1986) pp. 3–46
[a4] J.L. Journé, "Calderón–Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón" , Springer (1983)
How to Cite This Entry:
Cauchy integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_integral&oldid=13156
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article