Mathematical analysis
This article is about a branch of mathematics. The words "mathematical analysis" are also used to mean: the process or result of modeling and analyzing a phenomenon using mathematical techniques in general.Analysis is a branch of
mathematics that depends upon the concepts of
limits and
convergence. It studies closely related topics such as
continuity,
integration,
differentiability and
transcendental functions. These topics are often studied in the context of
real numbers,
complex numbers, and their
functions. However, they can also be defined and studied in any
space of mathematical objects that is equipped with a definition of "nearness" (a
topological space) or more specifically "distance" (a
metric space). Mathematical analysis has its beginnings in the rigorous formulation of
calculus.
Greek mathematicians such as
Eudoxus and
Archimedes made informal use of the concepts of limits and convergence when they used the
method of exhaustion to compute the area and volume of regions and solids.
In
India, the
12th century mathematician
Bhaskara conceived of
differential calculus, and gave examples of the
derivative and
differential coefficient, along with a statement of what is now known as
Rolle's theorem. In the
14th century, mathematical analysis originated with
Madhava in
South India, who developed the fundamental ideas of the
infinite series expansion of a function, the
power series, the
Taylor series, and the rational approximation of an infinite series. He developed the Taylor series of the
trigonometric functions of
sine,
cosine,
tangent and
arctangent, and estimated the magnitude of the error terms created by truncating these series. He also developed infinite
continued fractions, term by term
integration, the Taylor series approximations of sine and cosine, and the power series of the
radius,
diameter,
circumference,
π, π/4 and angle
θ. His followers at the
Kerala School further expanded his works, upto the
16th century.
Mathematical analysis in Europe began in the
17th century, with the
possibly independent invention of calculus by
Newton and
Leibniz. In the 17th and
18th centuries, analysis topics such as the
calculus of variations,
ordinary and
partial differential equations,
Fourier analysis and
generating functions were developed mostly in applied work. Calculus techniques were applied successfully to approximate
discrete problems by continuous ones.
All through the 18th century the definition of the concept of
function was a subject of debate among mathematicians. In the
19th century,
Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of the
Cauchy sequence. He also started the formal theory of
complex analysis.
Poisson,
Liouville,
Fourier and others studied partial differential equations and
harmonic analysis.
In the middle of the century
Riemann introduced his theory of
integration. The last third of the 19th century saw the arithmetization of analysis by
Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of
limit. Then, mathematicians started worrying that they were assuming the existence of a
continuum of
real numbers without proof.
Dedekind then constructed the real numbers by
Dedekind cuts. Around that time, the attempts to refine the
theorems of
Riemann integration led to the study of the "size" of the set of
discontinuities of real functions.
Also, "
monsters" (
nowhere continuous functions, continuous but nowhere differentiable functions,
space-filling curves) began to be created. In this context,
Jordan developed his theory of
measure,
Cantor developed what is now called
naive set theory, and
Baire proved the
Baire category theorem. In the early
20th century, calculus was formalized using
axiomatic set theory.
Lebesgue solved the problem of measure, and
Hilbert introduced
Hilbert spaces to solve
integral equations. The idea of
normed vector space was in the air, and in the
1920s Banach created
functional analysis.
Mathematical analysis includes the following subfields:
*
Real analysis, the
rigorous study of
derivatives and
integrals of functions of real variables. This includes the study of
sequences and their
limits,
series, and
measures.
*
Functional analysis studies spaces of functions and introduces concepts such as
Banach spaces and
Hilbert spaces.
*
Harmonic analysis deals with
Fourier series and their abstractions.
*
Complex analysis, the study of functions from the
complex plane to the complex plane which are complex differentiable.
*
p-adic analysis, the study of analysis within the context of
p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
*
Non-standard analysis, which investigates the
hyperreal numbers and their functions and gives a
rigorous treatment of
infinitesimals and infinitely large numbers. It is normally classed as
model theory.
Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called
hard analysis; it also naturally refers to the more traditional topics. The study of
differential equations is now shared with other fields such as
dynamical systems, though the overlap with 'straight' analysis is large.
*Nikol'skii, S.M.,
"Mathematical analysis", in
Encyclopaedia of Mathematics, Michiel Hazewinkel (editor), Springer-Verlag (2002). ISBN 1402006098.
