In the broad sense of the term, the theory of functions defined on some set of points
in the complex plane
(functions of a single complex variable) or on a set of points
of a complex Euclidean space
,
(functions of several complex variables). In the narrow sense of
the term, the theory of function of a complex
variable is the theory of analytic functions (cf.
Analytic function)
of one or several complex variables.
As an independent discipline, the theory of functions of a complex variable
took shape in about the middle of the
19th century
as the theory of analytic functions. The fundamental
work here was that of
A.L. Cauchy,
K. Weierstrass
and
B. Riemann,
who
approached the development of the theory from various (different) points of view.
According to Weierstrass, a function
is called
analytic
(or
holomorphic)
in a domain
if it can be expanded in a power series in a neighbourhood of each point
:
in the case of several complex variables, when
,
,
the series
(1)
is interpreted as a multiple power series. To
define an analytic function it is even sufficient that the convergent
series
(1)
be given in a neighbourhood of a single point
,
for its values at another point
and the corresponding series can be determined by the process of
analytic continuation
along various paths in the complex plane
(or in
,
)
joining
and
.
In the course of analytic continuation one may come across singular points (cf.
Singular point),
to which it is impossible to carry out analytic continuation
along any path. These singular points determine the general behaviour of
an analytic function in the sense that if two paths
and
joining the same fixed points
and
are not homotopic, that is, if it is impossible to deform
continuously into
without thereby passing through any singular point, then the values of the function
obtained by analytic continuation along
and
may turn out to be different. Consequently, the
complete analytic function
obtained by analytic continuation of an initial element
(1)
along all possible paths may turn out to be
multiple-valued in its natural domain of definition in
(or in
,
).
Examples of this are the functions
or
.
One can escape from this multiple-valuedness by forbidding
analytic continuation along certain paths, by constructing so-called
cuts
in the complex plane, and by distinguishing
single-valued branches of an analytic function (cf.
Branch of an analytic function).
But the most perfect method of converting a multiple-valued function into a single-valued one
consists in regarding it not as a function of a point of
the complex plane, but as a function of a point of a
Riemann surface,
consisting of several sheets that cover the complex plane, and joined to one
another in a certain way. In the case of several
variables, instead of a Riemann surface there arises a
Riemannian domain,
a multiple-sheeted cover of
,
.
In his construction of the theory of analytic
functions, Cauchy started from the concept of
monogeneity.
He called a function
,
,
monogenic
if it has a
monodromic
(that is, single-valued and continuous, except for poles)
derivative
everywhere in
.
Extending this concept somewhat, by a monogenic function
on a subset
one usually means a (single-valued) function for which there exists at all points
a
derivative with respect to
,
Monogeneity in the sense of Cauchy is the same as analyticity when
.
Cauchy developed the theory of integration of analytic
functions, proved the important theorem on residues (cf.
Residue of an analytic function),
the
Cauchy integral theorem,
and introduced the concept of the
Cauchy integral:
which expresses the value of an analytic function
in terms of its values on any closed contour
surrounding
and not containing any singular points of
inside or on
.
As the simplest integral representation of analytic functions, the concept of
the Cauchy integral can also be retained for functions of several variables.
If one introduces complex variables
,
,
one can describe any function of two variables
and
,
,
as a function of
and
.
The
Cauchy–Riemann conditions,
which pick out those among such functions that are analytic, demand that the functions
be differentiable with respect to both variables
,
while everywhere in
the equation
must hold, or, in full,
,
.
The conditions
(4)
mean that the real and imaginary parts
and
of an analytic function must be
conjugate harmonic functions.
In the case of analytic functions of several complex variables,
the conditions
(4)
must be satisfied with respect to all the variables
,
.
For Riemann, the most important thing was the circumstance that an analytic function
,
as picked out by the conditions
(4),
effects, under certain conditions, a
conformal mapping
of
onto some other domain in the plane of the complex variable
.
The connection between analytic functions and conformal mappings opens a
way to solving a number of problems in mathematical physics.
The subsequent development of the theory of functions of a complex variable
has been and still is above all a deepening and
extension of the theory of analytic functions (see, for example,
Boundary value problems of analytic function theory;
Boundary properties of analytic functions;
Uniqueness properties of analytic functions;
Integral representation of an analytic function;
Meromorphic function;
Multivalent function;
Univalent function;
Entire function).
Problems, related to analytic functions, of approximation and interpolation of functions have
an important significance. In these it turns out that in the
theory of analytic functions of several variables the specific nature and difficulty
of the problems are such that they only yield a solution when
one invokes the most modern methods of algebra, topology and analysis.
The boundary properties of holomorphic functions, in particular
of the integral of Cauchy type (see
Cauchy integral)
obtained from
(3)
when the values of
on the contour
are given totally arbitrarily, are of great theoretical and practical
significance, as are multi-dimensional analogues of
this and other integral representations.
Generalized analytic functions (cf.
Generalized analytic function),
which are important for applications, are obtained in their
simplest form as solutions of an equation generalizing
(4):
Their main properties (in the case of a
single variable) have been investigated in fair detail.
The study of quasi-conformal mappings (cf.
Quasi-conformal mapping)
is of great significance for the theory of
analytic functions itself (in particular, for the theory
of Riemann surfaces) and for its applications.
A theory of abstract analytic functions (cf.
Abstract analytic function)
with values in various vector spaces has also been developed.