A Cauchy integral is an integral with the
Cauchy kernel,
expressing the values of a regular analytic function
in the interior of a contour
in terms of its values on
.
More precisely: Let
be a regular analytic function of the complex variable
in a domain
and let
be a closed piecewise-smooth Jordan curve lying in
together with its interior
;
it is assumed that
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:
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
;
and 2) their integrands have the form
where
and
is a regular analytic function on
and in the interior of
.
If
(the complement to
)
in the Cauchy integral, i.e. if
lies outside
,
then, provided that the conditions 1) and 2) remain valid,
In particular, if
is the circle of radius
centred at a point
,
i.e.
then
(1)
implies that
i.e. the value of
at any point
is equal to the arithmetic average of its values on any sufficiently small circle
centred at
.
Formula
(1)
enables one to prove all other elementary properties of analytic functions.
On the other hand, if
is a regular analytic function in the infinite domain
(the exterior of the closed curve
)
and on
,
and if one defines
then the following formula, known as the
Cauchy integral formula for an infinite domain,
is valid:
Now let
be some (not necessarily closed) piecewise-smooth curve in the finite plane,
,
let
be a continuous complex function on
and let
be a point not on
.
The term
integral of Cauchy type
is applied to the following generalization of the Cauchy integral:
The function
is called the
density of the integral of Cauchy type.
Elementary properties of integrals of Cauchy type are:
1)
is a regular analytic function of
in any domain not containing points of
.
2)
The derivatives
are given by the formulas
3)
is regular at infinity, with
,
as
.
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
,
and to find analytic expressions for these values. The Cauchy integral
(1)
is equal to
everywhere in the interior of
and vanishes identically outside
.
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
is approached from the left (i.e. from its interior), the function
has boundary values
,
and if these values are assumed on
it is continuous from the left on
at each point
;
as
is approached from the right (i.e. from its exterior), then
has boundary values zero, i.e.
,
and if these values are assumed on
it is continuous from the right on
at each point
.
Thus, for a Cauchy integral
 |
For an integral of Cauchy type of general form
the matter is somewhat more complicated. Suppose that the equation of the curve
is
,
where
denotes the arc length reckoned from some fixed point, let
be an arbitrary fixed point on
and let
be the part of
that remains after the smaller of the arcs with end points
and
is deleted from
.
If the limit
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
is smooth in a neighbourhood of a point
distinct from the end points of
and if the density
satisfies a Hölder condition
Under these conditions there also exist boundary values, and these are given by the
Sokhotskii formulas:
and the functions
and
are continuous in a neighbourhood of
from the left and right, respectively, of
.
In the case of a Cauchy integral, the singular integral is equal to
An equivalent form of
(5)
is
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
be an arbitrary rectifiable curve of length
;
for simplicity it is assumed that
is closed. Let
be the angle between the direction of the
-axis
and the tangent to
at the point
,
regarded as a function of the arc length
,
and let
be a complex function of
of bounded variation on
.
The expression
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
.
If
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:
where
.
Let
be a point of
at which there exists a well-defined tangent, inclined to the
-axis
at an angle
;
such points exist almost-everywhere on a rectifiable curve. Let
be the point on the straight line passing through
and inclined to the normal at an angle
,
at a distance
,
i.e.
.
The difference between the integral of Cauchy–Stieltjes type
(8)
and the integral over
,
is defined at all points
where the tangent is defined, i.e. almost-everywhere on
.
An important proposition in the theory of integrals of Cauchy–Stieltjes type is
Privalov's fundamental lemma:
The limit
exists for all points
,
with the possible exception of a point set of measure zero on
,
independent of
;
the convergence is uniform in
in any angle
,
.
If the singular integral exists almost-everywhere on
,
then the integral of Cauchy–Stieltjes type has angular boundary values
almost-everywhere on
and these satisfy the
Sokhotskii formulas:
The converse is also true: If an integral of
Cauchy–Stieltjes type has angular boundary values from both inside and outside
,
almost-everywhere on
,
then the singular integral exists and formulas
(10)
are valid almost-everywhere on
.
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
,
an integral of Cauchy–Stieltjes type, even when it
has angular boundary values, is no longer
necessarily a continuous function in a neighbourhood of
from the left or right of
.
It is known, for example, that an integral of
Cauchy–Lebesgue type
(9)
is continuous in the closed domain
bounded by the rectifiable contour
,
provided one additionally assumes that the density
satisfies a Lipschitz condition on
:
One says that an integral of Cauchy–Lebesgue type
(9)
becomes a Cauchy integral
in the sense of Lebesgue, if the angular boundary values
from the inside of
coincide with
almost-everywhere on
,
i.e.
almost-everywhere on
.
In this context the
Golubev–Privalov theorem
holds: A summable function
on
represents the angular boundary values of some Cauchy integral from the inside of
if and only if all its moments vanish:
If the analogous conditions
are satisfied, then the integral of Cauchy–Stieltjes type
(8)
becomes a
Cauchy–Stieltjes integral:
i.e. the angular boundary values
from the inside of
coincide with the derivative
almost-everywhere on
,
or, stated differently, the angular boundary values
from the outside of
vanish almost-everywhere on
.
Conditions
(13)
immediately imply that the function
is absolutely continuous on
and, consequently, in this case the Cauchy–Stieltjes integral
(14)
is in fact a
Cauchy–Lebesgue integral
with density
.
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
bounded by a closed rectifiable curve
,
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
,
,
and
see
Boundary properties of analytic functions.
In the simplest case, when
is the unit disc and
is the unit circle, an integral of Cauchy–Stieltjes
type, which in this case has the form
always represents a function of class
,
.
The converse is false: The set of functions of classes
,
,
is more extensive than the set of functions representable in
the form
(15).
On the other hand, the set of functions representable in
by a Cauchy–Stieltjes or a Cauchy integral is identical with the class
.
In the case of an arbitrary simply-connected domain
bounded by a rectifiable curve
,
the class of functions representable in
by a Cauchy–Stieltjes or a Cauchy integral is identical with the
Smirnov class
(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
be an arbitrary (non-analytic) function of class
in a finite closed domain
bounded by a piecewise-smooth Jordan curve
.
The term
Cauchy integral formula
is sometimes applied also to the following generalization of the classical formula
(1):
where
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
be a regular analytic function of several complex variables
in a closed polydisc
,
.
Then, at each point of
,
is representable by a
multiple Cauchy integral:
where
is the distinguished boundary of the polydisc,
,
,
.
Formula
(17)
yields a simple analogue of the Cauchy integral for a circle
,
but when
the integration in
(17)
extends not over the entire boundary of
the polydisc but only over its distinguished boundary. In general, let
be a polycircular domain in
—
a product of simply-connected plane domains
with smooth boundaries
;
let
be the distinguished boundary of
,
which is a smooth
-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
the theory is concerned mainly with boundary
properties of integral representations other than
(17).