The representation of an analytic function by an integral depending
on a parameter. Integral representations of analytic functions arose in the
early stages of development of function theory and mathematical analysis in
general as a suitable apparatus for the explicit representation of
analytic solutions of differential equations, for the investigation of the asymptotics
of these solutions and for their analytic continuation. Somewhat later
integral representations of analytic functions found application in the solution of
boundary value problems of analytic function theory
and singular integral equations (cf.
Singular integral equation),
in the study of interior and
boundary properties of analytic functions
of various classes, as well as in the solution of other
problems of mathematical analysis. In the process of development of function
theory, the study of properties of the most important
individual integral representations of analytic functions
established an independent chapter in function theory (see, for example,
Cauchy integral;
Poisson integral;
Schwarz integral).
A wide class of integral representations of analytic functions, used
for obtaining and studying analytic solutions of differential
equations, can be described by the general formula
where
is the
kernel of the integral representation,
is its
density
and
is a contour (or system of contours) in the complex plane in which both variables
and
vary. An appropriate and, as far as possible, simplest
solution of the three interrelated questions on the choice of the kernel
,
the density
and the contour
for the representation of a given function
(or given class of functions) is the determining factor
from the point of view of the successful application of the
method of integral representations of analytic functions.
In turn, the properties of the representation
(1)
depend essentially on whether the kernel
is an entire function of the complex variables
,
or whether it is singular, that is, has some singularities.
In general, the kernel of an integral representation of an analytic function
does not have to be an analytic function of the variables
;
the analyticity of
may be ensured by specific properties of the density. Nor
is it necessary, in general, that formula
(1)
be a formula with
a single integration; there are types of integral representations
of analytic functions in which iterated integrals are used.
The general scheme for obtaining integral representations of special functions
that are solutions of certain ordinary differential equations
reduces in the main to the following. By an appropriate choice of the kernel
,
most often non-singular, the following formula for the action of the operator
should hold:
that is, the kernel must, in turn, as far
as possible satisfy a simple partial differential equation
,
thus allowing successive integration by parts with the aim of recovering the original
form of the kernel and a transfer to the action of the adjoint operator
thus obtained on the density
.
Having obtained a formula of the type
(2),
one selects a fairly simple density
satisfying the adjoint equation
,
and a contour
guaranteeing that the term
vanishes. Here one must bear in mind that the choice of the contour
determines a particular solution of the original equation
.
The following kernels are used most often:
sometimes called the
Laplace–Fourier kernel,
the
Mellin kernel
and the
Euler kernel,
respectively. Various changes of variables lead to modified
forms of the kernels. In the form written above,
integral representations of analytic functions are closely related
with the method of integral transforms (cf.
Integral transform).
In this way, for example, a well-known
integral representation of the Bessel functions
is obtained:
where the contour
takes the form of a figure eight containing the points
and
.
The representation
(3)
is distinctive in that, on the one hand, its density
is considerably simpler than the transcendental functions
being represented, and on the other hand it
enables one to survey fairly simply the properties of the functions
,
in particular, to study their asymptotics.
An appropriate modification of the contour
enables one to obtain analytic continuations, in
other words, to obtain integral representations that can be
used throughout the entire domain of existence. For example, the
Euler integral of the second kind,
represents the
gamma-function
for
,
and if the contour of integration
is taken to be a loop
(
Fig. a), then one obtains the integral representation
which holds for all
except for the points
at which
has simple poles.
Figure: i051620a
Figure: i051620b
Similarly, there is an analytic continuation of the
Euler integral of the first kind,
expressing the
beta-function
for
.
It is obtained by going over a contour in the form of a double loop
(
Fig. b).
One has studied integral representations of special functions (see
[1],
[2])
as well as integral representations of very wide
classes of functions, in connection with integral transforms
[7].
Of universal character in the theory of analytic functions is the singular
Cauchy kernel
and the corresponding integral representation, the
Cauchy integral
This integral representation expresses the values of a single-valued analytic function
in a domain
bounded by a simple closed contour
(or a system of such contours), for example, in the case when the function
is continuous in the closed domain
;
in the complementary domain
,
,
the integral on the right-hand side of
(4)
vanishes identically. The fundamental role of the representation
(4)
in
the theory of analytic functions is due to the
fact that the Cauchy integral is the convolution of
with the fundamental solution
of the
Cauchy–Riemann operator
Therefore all the fundamental properties of the analytic function can be obtained
from the representation
(4).
From the point of view of
general properties of integral representations of analytic functions, the Cauchy
integral is distinguished by the especially simple structure of
the kernel and by the fact that the density
coincides with the values of the represented function on the contour
.
This last property remains true if under the integral sign in
(4),
the Cauchy kernel
is replaced by any function
that is a single-valued analytic function in
in a closed domain
and has a simple pole at the point
with residue 1. The Cauchy kernel is the simplest of such functions
,
but the above method of choosing the kernel in the Cauchy
integral is often used in the solution of boundary value problems.
The fact that the density
coincides with the boundary value
of the analytic function
is in essence merely a form of expressing
the property of analyticity. In using the integral representation
(4)
with an a priori arbitrarily given density
,
an integral of Cauchy type is obtained in which the relationship between the density
and the boundary values is expressed in a considerably more complex
way in terms of a singular integral over the contour
.
In boundary value problems of analytic function theory the
role of Cauchy-type integrals and their modifications is exceptionally
important for the solution of singular integral equations (see
[5],
[6]).
In studying the interior and boundary properties of analytic functions
of various classes, more general integral representations of analytic functions than
(1)
are used, in the form of integrals depending on a parameter,
with respect to a Borel
boundary measure
which is in general complex, concentrated on the contour
bounding
and can be expressed by some procedure or other in terms of the function
being represented.
For example, all functions
that are regular in the unit disc
and have positive real part,
,
can be characterized by their representation in the
Herglotz formula
the idea of which essentially goes back to the
Schwarz integral.
Here
is an arbitrary positive measure concentrated on the circle
.
In the theory of univalent functions, a wide variety of
other integral representations of analytic functions of type
(5)
find
important applications; they are also known under
the name of parametric representations or
structural formulas
(cf.
Parametric representation of univalent functions).
Thus, for the class of
typically-real functions
in the disc
(that is, functions
for which
for
and
for
)
a typical representation is
where
is an arbitrary measure concentrated on the circle
and normalized by the condition
.
One also often uses a modification of the representation
(5)
in the form
where
is a specially chosen function, as simple as possible, for example
.
By regarding the measure
as a functional, the representation
(5)
can be interpreted as the value
of the functional
on the kernel
.
Consequently, a development of the method of integral representations of
analytic functions is the analytic representation of generalized functions
as the value of
on the kernel
:
Here, in the complement of the support of
the function
is analytic (the kernel
is assumed to be analytic in
for
).
Representations of the form
(6)
find important applications in mathematical physics (see
[10],
[11]).
In the theory of analytic functions
of several complex variables
,
,
integral representations in their simplest form can be expressed by a general formula:
where
is a density, somehow related to
,
is a differential form in the variables
,
,
whose coefficients depend on the parameters
,
,
and the integration is carried out over the entire boundary
of the domain of definition
of
or over some part of it. Representations in the form of a linear
combination of integrals of the type
(7)
have also been used. For example, a function
holomorphic in a polydisc-type domain
and continuous in the closure
is representable throughout
by a Cauchy integral:
where the differential form
has a very simple form:
and the integration is carried out over the
skeleton
of
(see
Bergman–Weil representation;
Leray formula;
Bochner–Martinelli representation formula,
and also
[8],
[9]).
As in the case of a single complex variable, the further
development of integral representations
(7)
are representations of the form:
or
expressing the analytic function
in some domain in the form of the value of a functional
on the kernel
or on the kernel-form
.
Here
is interpreted, respectively, as a generalized function on a
definite function space or as a current on a definite space of differential forms.