A function that can be locally represented by power series. The exceptional
importance of the class of analytic functions is due
to the following reasons. First, the class is sufficiently
large;
it includes the majority of functions which are encountered in the principal
problems of mathematics and its applications to science and
technology. Secondly, the class of analytic functions is
closed
with respect to the fundamental operations of
arithmetic, algebra and analysis. Finally, an important
property of an analytic function is its
uniqueness:
Each analytic function is an
"organically connected whole" ,
which represents
a
"unique"
function throughout its natural domain of existence. This property, which
in the
18th century
was considered as inseparable from the very notion
of a function, became of fundamental significance after a function had come to
be regarded, in the first half of the
19th century,
as
an arbitrary correspondence. The theory of analytic functions originated in the
19th century,
mainly due to the work of
A.L. Cauchy,
B. Riemann
and
K. Weierstrass.
The
"transition to the complex domain"
had a decisive effect on this
theory. The theory of analytic functions was constructed as the theory of functions
of a complex variable; at present (the
1970's) the theory of analytic functions
forms the main subject of the general theory of functions of a complex variable.
There are different approaches to the concept of analyticity.
One definition, which was originally proposed by Cauchy, and
was considerably advanced by Riemann, is based on a
structural property
of the function
— the existence of a derivative with respect to the complex variable,
i.e. its
complex differentiability.
This approach is closely connected with
geometric ideas.
Another approach, which was systematically developed by Weierstrass,
is based on the possibility of representing functions by
power series; it is thus connected with the
analytic apparatus
by means of which a function can be expressed. A basic fact
of the theory of analytic functions is the identity of the corresponding
classes of functions in an arbitrary domain of the complex plane.
Exact definitions are given below. Let
be a domain in the complex plane
.
If to each point
there has been assigned some complex number
,
then one says that on
a (single-valued) function
of the complex variable
has been defined and one writes:
,
(or
).
The function
may be regarded as a complex function of two real variables
and
,
defined in the domain
(where
is the Euclidean plane). To define such a
function is tantamount to defining two real functions
Having fixed a point
,
one gives
the increment
(such that
)
and considers the corresponding increment of the function
:
If
as
,
or in other words, if
exists, the function
is said to be
differentiable
(in the sense of complex analysis or in the sense of
)
at
;
is the derivative of
at
,
and
is its
differential
at that point. A function
which is differentiable at every point of
is called
differentiable in the domain
.
One may compare the concepts of differentiability of
considered as a function of two variables (in the sense of
)
and in the sense of
.
In the former case the differential
has the form
where
are the partial derivatives of
.
Passing from the independent variables
to the variables
,
which may formally be considered as new independent variables,
related to the old ones by the equations
,
(from this point of view, the function
may also be written as
)
and expressing
and
in terms of
and
according to the usual rules of differential calculus, one can write
in its complex form:
where
are the (formal) derivatives of
with respect to
and
,
respectively. It is seen, accordingly, that
is differentiable in the sense of
if and only if it is differentiable in the sense of
and if the equation
is satisfied, which in expanded form may be written as
If
is differentiable in the sense of
in
,
the latter relations are satisfied at all point of the domain; they are called the
Cauchy–Riemann equations.
These equations occurred already in the
18th century
in
J.L. d'Alembert's
and
L. Euler's
studies on functions of a complex variable. The initial definition
may now be rendered more precise as follows. A function
,
defined in a domain
,
is said to be
holomorphic
(analytic)
at a point
if there exists a neighbourhood of this point in which
may be represented by a power series:
If this requirement is satisfied at every point
of
,
the function
is said to be
holomorphic
(analytic)
in the domain
.
A function
which is holomorphic at a point
is differentiable at that point. In addition, the sum of a convergent power
series has derivatives of all orders (is infinitely
differentiable) with respect to the complex variable
;
the coefficients of the series may be expressed in terms of the derivatives of
at
by the formulas
.
The power series, written in the form
is known as the
Taylor series
of
at
.
Thus, holomorphy of a function
in a domain
means that it is infinitely differentiable at any point in
and that its Taylor series converges to it in some neighbourhood of this point.
On the other hand, the following noteworthy fact is established
in the theory of analytic functions: If a function
is differentiable in a domain
,
it is holomorphic in this domain (for a single point this statement is not true:
is differentiable at
,
but is nowhere holomorphic). Thus, the concepts of
complex differentiability and holomorphy of a function
in a domain
are identical; each one of the following properties of a function
in a domain
—
differentiability in the sense of
,
differentiability in the sense of
together with satisfaction of the Cauchy–Riemann equations,
holomorphy — may serve as definition of
analyticity
of
in this domain.
One other characteristic of an analytic function is connected with
the notion of an integral. The integral of a function
along an (oriented rectifiable) curve
:
,
,
may be defined by the formula:
or by means of a curvilinear integral:
A key result in the theory of analytic functions is
Cauchy's integral theorem:
If
is an analytic function in a domain
,
then
for any closed curve
bounding a domain inside
.
The converse result
(Morera's theorem)
is also true: If
is continuous in a domain
and if
for any such curve
,
then
is an analytic function in
.
In particular, in a simply-connected domain, those and only those continuous functions
are analytic, whose integral along any closed curve
is zero (or, which is the same thing, the integral along any curve
connecting two arbitrary points
does depend only on the points
and
themselves and not on the shape of the curve). This characterization
of analytic functions forms the basis of many of their applications.
Cauchy's integral theorem yields
Cauchy's integral formula,
which expresses the values of an analytic function inside a
domain in terms of its values on the boundary:
Here,
is a domain whose boundary
consists of a finite number of non-intersecting rectifiable curves (the orientation of
is assumed to be positive with respect to
),
and
is a function which is analytic in some domain
.
This formula makes it possible, in particular, to reduce the study of many
problems connected with analytic functions to the corresponding
problems for a very simple function — the
Cauchy kernel
,
,
.
For more details see
Integral representation of an analytic function.
A very important property of analytic functions is expressed by the following
uniqueness theorem:
Two functions which are analytic in a domain
and which coincide on some set with a limit point in
,
coincide throughout
(are identical). In particular, an analytic function
,
,
which is not identically zero can only have isolated zeros in
.
If, in addition,
is a zero of
,
then one has, in some neighbourhood
of
,
,
where
is a natural number (called the
multiplicity of the zero
of
at
),
while
is a analytic function in
.
An important role in the theory of analytic functions
is played by the points at which the function is not analytic — the so-called
singular points of the analytic function.
Here, only
isolated singular points
of (single-valued)
analytic functions are considered; for more details cf.
Singular point.
If
is an analytic function in an annulus of the form
,
it may be expanded there in a
Laurent series
which contains, as a rule, not only positive but also negative powers of
.
If there are no terms with negative powers in the series
(
for
),
is called a
regular point
of
(a removable singular point). At a regular point there also exists a finite limit
Putting
,
one obtains an analytic function in the whole disc
.
If the Laurent series of the function contains
only a finite number of terms with negative powers of
:
the point
is called a
pole
of
(of
multiplicity
);
a pole
is characterized by
The function
has a pole of multiplicity
at the point
if and only if the function
has a zero of multiplicity
at this point. If the Laurent series contains an infinite number of negative powers of
(
for an infinite set of negative indices
),
then
is called an
essential singular point;
at such points there is no finite and no infinite limit for
.
The coefficient
in the Laurent series for
with centre at the isolated singular point
is called the
residue
of
at
:
It can be defined by the formula
where
and
is sufficiently small (so that the disc
does not contain singular points of
other than
).
The important role of
residues
is made clear by the following theorem: If
is an analytic function in a domain
,
except for some set of isolated singular points, if
is a contour bounding a domain
and not passing through any singular points of
,
and if
are all the singular points of
inside
,
then
This theorem is an effective tool in calculating integrals. See also
Residue of an analytic function.
The sum of the terms of the Laurent series for
at
corresponding to the negative indices
,
is known as the
principal part
of the Laurent series (or of the function
)
at the point
.
This principal part determines the nature of the singularity of
at
.
Functions which are representable as a quotient of
two functions that are holomorphic in a domain
are called
meromorphic in the domain
.
A function which is meromorphic in a domain is holomorphic in that domain, except
possibly at a finite or countable set of poles; at the poles the
values of a meromorphic function are considered to be infinite. If
such values are allowed, then meromorphic functions in a domain
may be defined as functions that in a neighbourhood of each point
can be represented by a Laurent series in
with a finite number of terms involving negative powers of
(depending on
)
in a neighbourhood of each point
.
Both holomorphic and meromorphic functions in a domain
are often designated as
analytic in the domain
.
In this a case holomorphic functions are also said to be
regular analytic
or simply
regular
functions.
The simplest class of analytic functions consists of the functions which
are holomorphic in the whole plane; such functions are called
entire functions.
Entire functions are represented by series
which are convergent in the whole plane. This class includes the polynomials in
,
the functions
etc.
Weierstrass' theorem
states that, for any sequence of complex numbers
,
without limit points in
,
there exists an entire function
that vanishes at the points
and only at these points (among the
there may be coincident points, to which correspond zeros of
of corresponding multiplicity). Here, the function
may be represented as a (generally infinite) product of entire
functions each one of which has only one zero.
Functions that are meromorphic in the plane (i.e. that
may be represented as quotients of entire functions) are called
meromorphic functions.
These include rational functions,
,
,
elliptic functions, etc.
According to
Mittag-Leffler's theorem,
for any sequence
,
without limit points in
,
there exists a meromorphic function
with poles at the points
and only at those points, such that its principal parts at the points
coincide with pre-assigned polynomials in
.
The function
may be represented as a (usually infinite) sum of meromorphic functions,
each one with a pole at a single point only.
Theorems on the existence of a holomorphic function with pre-assigned
zeros and of meromorphic functions with pre-assigned poles and
principal parts are also valid for an arbitrary domain
.
In the study of analytic functions the related
geometric notions are also of importance. If
is an analytic function, the image
of the domain
is also a domain
(principle of preservation of domains).
,
If the mapping
preserves angles at
both in value and in sign, i.e. it is conformal. Thus, there
exists a close connection between analyticity and the important geometric notion of
conformal mapping.
If
is an analytic function in
and
for
(such functions are called
univalent),
then
in
and
defines a one-to-one and conformal mapping of the domain
onto the domain
.
Riemann's theorem,
which is the fundamental theorem in the theory of conformal mappings,
says that on any simply-connected domain whose boundary contains more than
one point there exist univalent analytic functions which conformally map
this domain onto a disc or a half-plane (cf.
Conformal mapping;
Univalent function).
The real and imaginary parts of a function
which is holomorphic in a domain
satisfy the
Laplace equation
in that domain:
i.e. they are harmonic functions (cf.
Harmonic function).
Two harmonic functions which are connected by the Cauchy–Riemann equations are called
conjugate.
In a simply-connected domain any harmonic function
has a conjugate function
and is thus the real part of some holomorphic function
in
.
The connections with conformal mappings and harmonic functions form the
basis of many applications of the theory of analytic functions.
A function
(
being an arbitrary set), is called
analytic at a point
if there exists a neighbourhood of this point such that
may be represented by a convergent power series
on the intersection of this neighbourhood with
.
The function
is called
analytic on the set
if it is analytic on some open set which contains
(or, more exactly, if there exist both an open set containing
and an analytic function
on this set which coincides with
on
).
For
open
sets the notion to analyticity coincides with the notion of differentiability with respect
to the set. However, this is not the case in general; in particular,
on the real line there exist functions which not only have a derivative,
but which are infinitely differentiable at every point and
are not analytic even at a single point of this line. The property of
connectedness
of the set
is necessary in order that the uniqueness theorem for analytic
functions holds. This is why analytic functions are usually considered in
domains,
i.e. on connected open sets.
All the preceding refers to
single-valued
analytic functions
,
considered in a given domain
(or on a given set
)
of the complex plane. In considering the possible extension of a function
,
as an analytic function, to a larger domain, one arrives at the concept of
the analytic function considered as a whole, i.e. throughout its whole natural domain
of existence. If the function is thus extended, its domain of
analyticity becomes larger, and may overlap itself, supplying new values of
the function at points in the plane where it already was
defined. Accordingly, an analytic function considered as a whole is generally
multi-valued.
Many problems in analysis (inversion of a function, the determination
of a primitive and the construction of an analytic function
with a given real part in multiply-connected domains (cf.
Multiply-connected domain),
the solution of algebraic equations with analytic coefficients,
etc.) require the study of multi-valued functions; such functions include
,
,
,
,
algebraic functions, etc. (cf.
Algebraic function).
A regular process which yields the
complete analytic function,
considered throughout its natural domain of existence, was
proposed by Weierstrass; it is known as
Weierstrass' method
of
analytic continuation.
The initial concept is that of an
element
of an analytic function, viz. a power series
with a non-zero radius of convergence. Such an element
:
defines a certain analytic function
on its disc of convergence
.
Let
be a point of
different from
.
Expanding
in a series with centre at
,
one obtains a new element
:
whose disc of convergence will be denoted by
.
In the intersection of
and
the series
converges to the same function as the series
.
If
extends beyond the boundary of
,
the series
defines the function determined by
on some set outside
(where the series
is divergent). In such a case the element
is called a direct analytic continuation of the element
.
Let
be a chain of elements in which
is a direct analytic continuation of
(
);
the element
is then said to be an analytic continuation of the element
(by means of the given chain of elements). When the centre of the disc
belongs to
it may happen that the element
is not a direct analytic continuation of the element
.
In such a case the sums of the series
and
will have different values in the intersection of
and
;
thus analytic continuation may lead to new values of the function inside
.
The totality of all elements which may be obtained by analytic continuation of an element
forms the
complete analytic function
(in the sense of Weierstrass) generated by
;
the union of their discs of convergence represents the (Weierstrass)
domain of existence
of this function. It follows from the uniqueness theorem for analytic
functions that an analytic function in the sense of
Weierstrass is completely determined by the given element
.
The initial element may be any other element
belonging to this function; the complete analytic function will not be affected.
A complete analytic function
,
considered as a function of the points in the plane belonging to its domain of existence
,
is generally multi-valued. In order to eliminate this feature,
is considered not as a function of the points in the plane domain
,
but rather as a function of the points on some multi-sheeted surface
(lying above
)
such that to each point of
there correspond as many points of the surface
(projecting onto the given point of
)
as there are different elements with centre at
this point in the complete analytic function
;
on the surface
the function
becomes single-valued. The idea of passing to such surfaces is
due to Riemann, and the surfaces themselves are known as
Riemann surfaces.
The abstract definition of the notion of a Riemann surface
has made it possible to replace the theory of multi-valued
analytic functions by the theory of single-valued
analytic functions on Riemann surfaces (cf.
Riemann surface).
Now fix a domain
belonging to the domain of existence
of the complete analytic function
,
and fix some element
of
with centre at a point in
.
The totality of all elements which may be obtained by analytic continuation of
by means of chains with centres belonging to
is called a
branch of the analytic function
.
A branch of a multi-valued analytic function may turn out
to be a single-valued analytic function in the domain
.
Thus, arbitrary branches of the functions
and
which correspond to an arbitrary simply-connected domain not containing
the point 0, are single-valued functions. The function
has exactly
different branches in such a domain, while
has an infinite set of such branches. The
selection of single-valued branches (using some cuts in the domain
of existence) and their study by the theory of single-valued analytic
functions constitute one of the principal methods
of studying specific multi-valued analytic functions.