Mathematical analysis is the branch of mathematics
dealing with limits
and related theories, such as differentiation,
integration, measure, infinite series, and
analytic functions.These theories are usually
studied in the context of real and complex
numbers and functions. Analysis evolved from
calculus, which involves the elementary concepts
and techniques of analysis.
Analysis may be distinguished from geometry;
however, it can be applied to any space of
mathematical objects that has a definition
of nearness (a topological space) or specific
distances between objects (a metric space).
== History ==
Mathematical analysis formally developed in
the 17th century during the Scientific Revolution,
but many of its ideas can be traced back to
earlier mathematicians. Early results in analysis
were implicitly present in the early days
of ancient Greek mathematics. For instance,
an infinite geometric sum is implicit in Zeno's
paradox of the dichotomy. Later, Greek mathematicians
such as Eudoxus and Archimedes made more explicit,
but 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. The explicit use of
infinitesimals appears in Archimedes' The
Method of Mechanical Theorems, a work rediscovered
in the 20th century. In Asia, the Chinese
mathematician Liu Hui used the method of exhaustion
in the 3rd century AD to find the area of
a circle. Zu Chongzhi established a method
that would later be called Cavalieri's principle
to find the volume of a sphere in the 5th
century. The Indian mathematician Bhāskara
II gave examples of the derivative and used
what is now known as Rolle's theorem in the
12th century.In the 14th century, Madhava
of Sangamagrama developed infinite series
expansions, like the power series and the
Taylor series, of functions such as sine,
cosine, tangent and arctangent. Alongside
his development of the Taylor series of the
trigonometric functions, he also estimated
the magnitude of the error terms created by
truncating these series and gave a rational
approximation of an infinite series. His followers
at the Kerala School of Astronomy and Mathematics
further expanded his works, up to the 16th
century.
The modern foundations of mathematical analysis
were established in 17th century Europe. Descartes
and Fermat independently developed analytic
geometry, and a few decades later Newton and
Leibniz independently developed infinitesimal
calculus, which grew, with the stimulus of
applied work that continued through the 18th
century, into analysis topics such as the
calculus of variations, ordinary and partial
differential equations, Fourier analysis,
and generating functions. During this period,
calculus techniques were applied to approximate
discrete problems by continuous ones.
In the 18th century, Euler introduced the
notion of mathematical function. Real analysis
began to emerge as an independent subject
when Bernard Bolzano introduced the modern
definition of continuity in 1816, but Bolzano's
work did not become widely known until the
1870s. In 1821, Cauchy began to put calculus
on a firm logical foundation by rejecting
the principle of the generality of algebra
widely used in earlier work, particularly
by Euler. Instead, Cauchy formulated calculus
in terms of geometric ideas and infinitesimals.
Thus, his definition of continuity required
an infinitesimal change in x to correspond
to an infinitesimal change in y. He also introduced
the concept of the Cauchy sequence, and started
the formal theory of complex analysis. Poisson,
Liouville, Fourier and others studied partial
differential equations and harmonic analysis.
The contributions of these mathematicians
and others, such as Weierstrass, developed
the (ε, δ)-definition of limit approach,
thus founding the modern field of mathematical
analysis.
In the middle of the 19th century Riemann
introduced his theory of integration. The
last third of the century saw the arithmetization
of analysis by Weierstrass, who thought that
geometric reasoning was inherently misleading,
and introduced the "epsilon-delta" 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,
in which irrational numbers are formally defined,
which serve to fill the "gaps" between rational
numbers, thereby creating a complete set:
the continuum of real numbers, which had already
been developed by Simon Stevin in terms of
decimal expansions. 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 investigated.
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 an 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.
== Important concepts ==
=== Metric spaces ===
In mathematics, a metric space is a set where
a notion of distance (called a metric) between
elements of the set is defined.
Much of analysis happens in some metric space;
the most commonly used are the real line,
the complex plane, Euclidean space, other
vector spaces, and the integers. Examples
of analysis without a metric include measure
theory (which describes size rather than distance)
and functional analysis (which studies topological
vector spaces that need not have any sense
of distance).
Formally, a metric space is an ordered pair
(
M
,
d
)
{\displaystyle (M,d)}
where
M
{\displaystyle M}
is a set and
d
{\displaystyle d}
is a metric on
M
{\displaystyle M}
, i.e., a function
d
:
M
×
M
→
R
{\displaystyle d\colon M\times M\rightarrow
\mathbb {R} }
such that for any
x
,
y
,
z
∈
M
{\displaystyle x,y,z\in M}
, the following holds:
d
(
x
,
y
)
=
0
{\displaystyle d(x,y)=0}
if and only if
x
=
y
{\displaystyle x=y}
(identity of indiscernibles),
d
(
x
,
y
)
=
d
(
y
,
x
)
{\displaystyle d(x,y)=d(y,x)}
(symmetry), and
d
(
x
,
z
)
≤
d
(
x
,
y
)
+
d
(
y
,
z
)
{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}
(triangle inequality).By taking the third
property and letting
z
=
x
{\displaystyle z=x}
, it can be shown that
d
(
x
,
y
)
≥
0
{\displaystyle d(x,y)\geq 0}
(non-negative).
=== Sequences and limits ===
A sequence is an ordered list. Like a set,
it contains members (also called elements,
or terms). Unlike a set, order matters, and
exactly the same elements can appear multiple
times at different positions in the sequence.
Most precisely, a sequence can be defined
as a function whose domain is a countable
totally ordered set, such as the natural numbers.
One of the most important properties of a
sequence is convergence. Informally, a sequence
converges if it has a limit. Continuing informally,
a (singly-infinite) sequence has a limit if
it approaches some point x, called the limit,
as n becomes very large. That is, for an abstract
sequence (an) (with n running from 1 to infinity
understood) the distance between an and x
approaches 0 as n → ∞, denoted
lim
n
→
∞
a
n
=
x
.
{\displaystyle \lim _{n\to \infty }a_{n}=x.}
== Main branches ==
=== 
Real analysis ===
Real analysis (traditionally, the theory of
functions of a real variable) is a branch
of mathematical analysis dealing with the
real numbers and real-valued functions of
a real variable. In particular, it deals with
the analytic properties of real functions
and sequences, including convergence and limits
of sequences of real numbers, the calculus
of the real numbers, and continuity, smoothness
and related properties of real-valued functions.
=== Complex analysis ===
Complex analysis, traditionally known as the
theory of functions of a complex variable,
is the branch of mathematical analysis that
investigates functions of complex numbers.
It is useful in many branches of mathematics,
including algebraic geometry, number theory,
applied mathematics; as well as in physics,
including hydrodynamics, thermodynamics, mechanical
engineering, electrical engineering, and particularly,
quantum field theory.
Complex analysis is particularly concerned
with the analytic functions of complex variables
(or, more generally, meromorphic functions).
Because the separate real and imaginary parts
of any analytic function must satisfy Laplace's
equation, complex analysis is widely applicable
to two-dimensional problems in physics.
=== Functional analysis ===
Functional analysis is a branch of mathematical
analysis, the core of which is formed by the
study of vector spaces endowed with some kind
of limit-related structure (e.g. inner product,
norm, topology, etc.) and the linear operators
acting upon these spaces and respecting these
structures in a suitable sense. The historical
roots of functional analysis lie in the study
of spaces of functions and the formulation
of properties of transformations of functions
such as the Fourier transform as transformations
defining continuous, unitary etc. operators
between function spaces. This point of view
turned out to be particularly useful for the
study of differential and integral equations.
=== Differential equations ===
A differential equation is a mathematical
equation for an unknown function of one or
several variables that relates the values
of the function itself and its derivatives
of various orders. Differential equations
play a prominent role in engineering, physics,
economics, biology, and other disciplines.
Differential equations arise in many areas
of science and technology, specifically whenever
a deterministic relation involving some continuously
varying quantities (modeled by functions)
and their rates of change in space or time
(expressed as derivatives) is known or postulated.
This is illustrated in classical mechanics,
where the motion of a body is described by
its position and velocity as the time value
varies. Newton's laws allow one (given the
position, velocity, acceleration and various
forces acting on the body) to express these
variables dynamically as a differential equation
for the unknown position of the body as a
function of time. In some cases, this differential
equation (called an equation of motion) may
be solved explicitly.
=== Measure theory ===
A measure on a set is a systematic way to
assign a number to each suitable subset of
that set, intuitively interpreted as its size.
In this sense, a measure is a generalization
of the concepts of length, area, and volume.
A particularly important example is the Lebesgue
measure on a Euclidean space, which assigns
the conventional length, area, and volume
of Euclidean geometry to suitable subsets
of the
n
{\displaystyle n}
-dimensional Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
. For instance, the Lebesgue measure of the
interval
[
0
,
1
]
{\displaystyle \left[0,1\right]}
in the real numbers is its length in the everyday
sense of the word – specifically, 1.
Technically, a measure is a function that
assigns a non-negative real number or +∞
to (certain) subsets of a set
X
{\displaystyle X}
. It must assign 0 to the empty set and be
(countably) additive: the measure of a 'large'
subset that can be decomposed into a finite
(or countable) number of 'smaller' disjoint
subsets, is the sum of the measures of the
"smaller" subsets. In general, if one wants
to associate a consistent size to each subset
of a given set while satisfying the other
axioms of a measure, one only finds trivial
examples like the counting measure. This problem
was resolved by defining measure only on a
sub-collection of all subsets; the so-called
measurable subsets, which are required to
form a
σ
{\displaystyle \sigma }
-algebra. This means that countable unions,
countable intersections and complements of
measurable subsets are measurable. Non-measurable
sets in a Euclidean space, on which the Lebesgue
measure cannot be defined consistently, are
necessarily complicated in the sense of being
badly mixed up with their complement. Indeed,
their existence is a non-trivial consequence
of the axiom of choice.
=== Numerical analysis ===
Numerical analysis is the study of algorithms
that use numerical approximation (as opposed
to general symbolic manipulations) for the
problems of mathematical analysis (as distinguished
from discrete mathematics).Modern numerical
analysis does not seek exact answers, because
exact answers are often impossible to obtain
in practice. Instead, much of numerical analysis
is concerned with obtaining approximate solutions
while maintaining reasonable bounds on errors.
Numerical analysis naturally finds applications
in all fields of engineering and the physical
sciences, but in the 21st century, the life
sciences and even the arts have adopted elements
of scientific computations. Ordinary differential
equations appear in celestial mechanics (planets,
stars and galaxies); numerical linear algebra
is important for data analysis; stochastic
differential equations and Markov chains are
essential in simulating living cells for medicine
and biology.
== Other topics ==
Calculus of variations deals with extremizing
functionals, as opposed to ordinary calculus
which deals with functions.
Harmonic analysis deals with the representation
of functions or signals as the superposition
of basic waves.
Geometric analysis involves the use of geometrical
methods in the study of partial differential
equations and the application of the theory
of partial differential equations to geometry.
Clifford analysis, the study of Clifford valued
functions that are annihilated by Dirac or
Dirac-like operators, termed in general as
monogenic or Clifford analytic functions.
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.
Computable analysis, the study of which parts
of analysis can be carried out in a computable
manner.
Stochastic calculus – analytical notions
developed for stochastic processes.
Set-valued analysis – applies ideas from
analysis and topology to set-valued functions.
Convex analysis, the study of convex sets
and functions.
Idempotent analysis – analysis in the context
of an idempotent semiring, where the lack
of an additive inverse is compensated somewhat
by the idempotent rule A + A = A.
Tropical analysis – analysis of the idempotent
semiring called the tropical semiring (or
max-plus algebra/min-plus algebra).
Non-Newtonian calculus, alternatives to the
classical calculus of Newton and Leibniz.
== Applications ==
Techniques from analysis are also found in
other areas such as:
=== Physical sciences ===
The vast majority of classical mechanics,
relativity, and quantum mechanics is based
on applied analysis, and differential equations
in particular. Examples of important differential
equations include Newton's second law, the
Schrödinger equation, and the Einstein field
equations.
Functional analysis is also a major factor
in quantum mechanics.
=== Signal processing ===
When processing signals, such as audio, radio
waves, light waves, seismic waves, and even
images, Fourier analysis can isolate individual
components of a compound waveform, concentrating
them for easier detection or removal. A large
family of signal processing techniques consist
of Fourier-transforming a signal, manipulating
the Fourier-transformed data in a simple way,
and reversing the transformation.
=== Other areas of mathematics ===
Techniques from analysis are used in many
areas of mathematics, including:
Analytic number theory
Analytic combinatorics
Continuous probability
Differential entropy in information theory
Differential games
Differential geometry, the application of
calculus to specific mathematical spaces known
as manifolds that possess a complicated internal
structure but behave in a simple manner locally.
Differentiable manifolds
Differential topology
Partial differential equations
== See also ==
Constructive analysis
History of calculus
Non-classical analysis
Paraconsistent logic
Smooth infinitesimal analysis
Timeline of calculus and mathematical analysis
== Notes
