The part of mathematics in which functions (cf.
Function)
and their generalizations are studied by the method of limits (cf.
Limit).
The concept of limit is closely connected with that of an
infinitesimal quantity, therefore it could be
said that mathematical analysis studies functions
and their generalizations by infinitesimal methods.
The name
"mathematical analysis"
is a short version of the
old name of this part of mathematics,
"infinitesimal analysisinfinitesimal analysis" ;
the latter more fully describes
the content, but even it is an abbreviation
(the name
"analysis by means of infinitesimalsanalysis by means of infinitesimals"
would characterize the
subject more precisely). In classical mathematical analysis
the objects of study (analysis) were first
and foremost functions.
"First and foremost"
because the development
of mathematical analysis has led to the possibility of
studying, by its methods, forms more
complicated than functions: functionals, operators, etc.
Everywhere in nature and technology one meets motions and processes
which are characterized by functions; the laws of natural phenomena
also are usually described by functions. Hence the objective importance
of mathematical analysis as a means of studying functions.
Mathematical analysis, in the broad sense of the term,
includes a very large part of mathematics. It includes
differential calculus;
integral calculus;
the theory of functions of a real variable (cf.
Functions of a real variable, theory of);
the theory of functions of a complex variable (cf.
Functions of a complex variable, theory of);
approximation theory;
the theory of ordinary differential equations (cf.
Differential equation, ordinary);
the theory of partial differential equations (cf.
Differential equation, partial);
the theory of integral equations (cf.
Integral equation);
differential geometry;
variational calculus;
functional analysis;
harmonic analysis;
and certain other mathematical disciplines. Modern
number theory
and
probability theory
use and develop methods of mathematical analysis.
Nevertheless, the term
"mathematical analysis"
is often used as a
name for the foundations of mathematical analysis, which
unifies the theory of real numbers (cf.
Real number),
the theory of limits, the theory of
series,
differential and integral calculus, and their immediate applications such as
the theory of maxima and minima, the theory of implicit functions (cf.
Implicit function),
Fourier series,
and Fourier integrals (cf.
Fourier integral).
Functions.
Mathematical analysis began with the definition of a function
by
N.I. Lobachevskii
and
P.G.L. Dirichlet.
If to each number
,
from some set
of numbers, is associated by some rule a number
,
then this defines a function
of one variable
.
A function of
variables,
is defined similarly, where
is a point of an
-dimensional
space; one also considers functions
of points
of some infinite-dimensional space. These, however, are usually called functionals.
Elementary functions.
In mathematical analysis the
elementary functions
are of fundamental importance. Basically, in practice, one operates
with the elementary functions and more complicated functions are approximated
by them. The elementary functions can be considered
not only for real but also for complex
;
then the conception of these functions becomes in some
sense, complete. In this connection an important branch of mathematics
has arisen, called the theory of functions of a
complex variable, or the theory of analytic functions (cf.
Analytic function).
Real numbers.
The concept of a function is essentially founded on the concept
of a real (rational or irrational) number. The
latter was finally formulated only at the end of
the
19th century.
In particular, it established a logically
irreproachable connection between numbers and points of a geometrical line, which
gave a formal foundation for the ideas of
R. Descartes
(mid
17th century),
who introduced into
mathematics rectangular coordinate systems and the representation of functions by graphs.
Limits.
In mathematical analysis a means of studying functions is the limit.
One distinguishes between the limit of a sequence and the limit
of a function. These concepts were finally formulated only in the
19th century;
however,
the idea of a limit had been studied by the ancient Greeks. It
suffices to say that
Archimedes
(3rd century B.C.)
was able to calculate the area
of a segment of a parabola by a process which one would call a limit transition (see
Exhaustion, method of).
Continuous functions.
An important class of functions studied in mathematical
analysis is formed by the continuous functions (cf.
Continuous function).
One of the possible definitions of this notion is: A function
,
of a variable
from an open interval
,
is called
continuous at the point
if
A function is
continuous on the open interval
if it is continuous at each of its points; its graph is then
a curve which is continuous in the everyday sense of the word.
Derivative and differential.
Among the continuous functions those having a
derivative
must be distinguished. The derivative of a function
at a point
is its rate of change at that point, that is, the limit
If
is the coordinate at the time
of a point moving along the coordinate axis, then
is its instantaneous velocity at the time
.
From the sign of
one can judge the nature of variation of
:
If
(
)
in an interval
,
then
is increasing (decreasing) on this interval. If a function attains
a local extremum (a maximum or a minimum) at
and has a derivative at this point, then the latter is equal to zero,
.
The equality
(1)
can be replaced by the equivalent equality
or
where
is an infinitesimal as
;
that is, if
has a derivative at
,
then its increment at this point decomposes into two terms. The first
is a linear function of
(is proportional to
),
the second term tends to zero more rapidly than
.
The quantity
(2)
is called the
differential
of the function corresponding to the increment
.
For small
it is possible to regard
as approximately equal to
:
These arguments about differentials are characteristic of mathematical analysis.
They have been extended to functions of several variables and to functionals.
For example, if a function
of
variables has continuous partial derivatives (cf.
Partial derivative)
at a point
,
then its increment
corresponding to increments
of the independent variables can be written in the form
where
as
,
that is, if all
.
Here the first term on the right-hand side in
(3)
is the differential
of
.
It depends linearly on
and the second term tends to zero more rapidly than
as
.
Suppose one is given a functional (see
Variational calculus)
extended over the class
of functions
having continuous derivatives on the closed interval
and satisfying the boundary conditions
,
,
where
and
are given numbers. Let, further,
be the class of functions
having continuous derivatives on
and such that
.
Obviously, if
and
,
then
.
In variational calculus it has been proved that under certain conditions on
the increment of
can be written in the form
as
,
where
and, thus, the second term on the right-hand side of
(4)
tends to zero more rapidly than
,
whereas the first term depends linearly on
.
The first term in
(4)
is called the
variation of the functional
and is denoted by
.
Integrals.
Side by side with the derivative, the integral has a
fundamental significance in mathematical analysis. One
distinguishes indefinite and definite integrals.
The indefinite integral is closely connected with primitive functions. A function
is called a
primitive function
of a function
on the interval
if, on this interval,
.
The definite (Riemann) integral of a function
on an interval
is the limit
as
;
here
and
are arbitrary.
If
is positive and continuous on
,
its integral on this segment is equal to the area of the figure bounded by the curve
,
the
-axis
and the lines
and
.
The class of Riemann-integrable functions contains all continuous functions on
and some discontinuous ones. But they are all necessarily
bounded. For a slowly-growing unbounded function, and also
for certain functions on unbounded intervals, the so-called
improper integral
has been introduced, requiring a double limit transition in its definition.
The concept of a Riemann integral of a function of one
variable can be extended to functions of several variables (see
Multiple integral).
On the other hand, the needs of mathematical analysis have led
to a generalization of the integral in quite another direction, in the form of the
Lebesgue integral
or, more generally, the
Lebesgue–Stieltjes integral.
Essential in the definition of these integrals
is the introduction for certain sets, called measurable, of their
measure and, on this foundation, the notion of a
measurable function. For measurable functions the Lebesgue–Stieltjes integral has been
introduced. In this connection a broad range of different measures has
been considered, together with the associated classes of measurable sets
and functions. This provides an opportunity to adapt this
or that integral to a definite concrete problem.
Newton–Leibniz formula.
There is a connection between derivatives and integrals, expressed by the
Newton–Leibniz formula
(theorem):
Here
is a continuous function on
and
is its primitive function.
Taylor's formulas and series.
Along with derivatives and integrals, the most important
ideas (research tools) in mathematical analysis are the
Taylor formula
and
Taylor series.
If a function
,
,
has continuous derivatives up to and including order
in a neighbourhood of a point
,
then it can be approximated in this neighbourhood by the polynomial
called its
Taylor polynomial
(of degree
),
in powers of
:
(Taylor's formula);
here the error of approximation,
tends to zero faster than
as
:
Thus, in a neighbourhood of
,
can be approximated to any degree of accuracy
by very simple functions (polynomials), which for their
calculation require only the arithmetic operations
of addition, subtraction and multiplication.
Of special importance are the so-called analytic functions in a fixed neighbourhood of
;
they have an infinite number of derivatives,
as
in the neighbourhood, and they may be represented by the infinite
Taylor series
Taylor expansions are also possible, under certain conditions,
for functions of several variables, functionals and operators.
Historical information.
Up to the
17th century
mathematical analysis was a collection
of solutions to disconnected particular problems; for example, in the integral
calculus, the problems of the calculation of the areas of figures, the
volumes of bodies with curved boundaries, the work done by a variable force,
etc. Each problem, or special group of problems, was solved by
its own method, sometimes complicated and tedious and
sometimes even brilliant (regarding the prehistory of mathematical analysis see
Infinitesimal calculus).
Mathematical analysis as a unified and systematic whole was put together in the
works of
I. Newton,
G. Leibniz,
L. Euler,
J.L. Lagrange,
and other scholars in the
17th century
and
18th century,
and its foundations, the theory of limits, was laid by
A.L. Cauchy
at the beginning of the
19th century.
A deep analysis
of the original ideas of mathematical analysis was connected with the development
in the
19th century
and
20th century
of set theory, measure theory and the theory
of functions of a real variable, and has led to a variety of generalizations.