mathematical analysis

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Analysis is the branch of
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
dealing with
limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
and related theories, such as
differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in marketing * Differentiated service, a service that varies with the identity o ...

, , measure,
sequences In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
, and
analytic function In mathematics, an analytic function is a function (mathematics), function that is locally given by a convergent series, convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are ...
s. These theories are usually studied in the context of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
and
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

numbers and
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
. Analysis evolved from
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from
geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...

; however, it can be applied to any
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gree ...
of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
s that has a definition of nearness (a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...
) or specific distances between objects (a
metric space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
).

# History

300px, Archimedes_used_the_method_of_exhaustion_to_compute_the_area.html" ;"title="method_of_exhaustion.html" ;"title="Archimedes used the method of exhaustion">Archimedes used the method of exhaustion to compute the area">method_of_exhaustion.html" ;"title="Archimedes used the method of exhaustion">Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit (mathematics), limit, one of the most basic concepts in mathematical analysis.

## Ancient

Mathematical analysis formally developed in the 17th century during the
Scientific Revolution The Scientific Revolution was a series of events that marked the emergence of modern science during the early modern period, when developments in History of mathematics#Mathematics during the Scientific Revolution, mathematics, History of phys ...

, 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 Greek mathematics refers to mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
. For instance, an infinite geometric sum is implicit in paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

made more explicit, but informal, use of the concepts of limits and convergence when they used the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of ...

to compute the area and volume of regions and solids. The explicit use of
infinitesimals In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
appears in Archimedes' ''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is considered one of the major surviving works of the ancient Greece, ...
'', a work rediscovered in the 20th century. In Asia, the Chinese mathematician
Liu Hui Liu Hui () was a Chinese mathematician and writer who lived in the state of Cao Wei Wei (220–266), also known as Cao Wei or Former Wei, was one of the three major states that competed for supremacy over China in the Three Kingdoms period ( ...
used the method of exhaustion in the 3rd century AD to find the area of a circle. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, [τέχνη], ''tiké [téchne]'', 'art' or 'cra ...
and geometric series, geometric series as early as the 4th century B.C. Bhadrabahu, Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 B.C. In Indian mathematics, particular instances of arithmetic series have been found to implicitly occur in Vedic Literature as early as 2000 B.C.

## Medieval

Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematics, Indian mathematician Bhāskara II gave examples of derivatives and used what is now known as Rolle's theorem. In the 14th century, Madhava of Sangamagrama developed series (mathematics), infinite series expansions, now called Taylor series, of functions such as Trigonometric functions, sine, Trigonometric functions, cosine, trigonometric functions, tangent and Inverse trigonometric functions, arctangent. Alongside his development of Taylor series of trigonometric functions, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century.

## Modern

### Foundations

The modern foundations of mathematical analysis were established in 17th century Europe. This began when Fermat and Descartes developed analytic geometry, which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. Descartes' publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis. It would be a few decades later that Isaac Newton, Newton and Gottfried Leibniz, 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 differential equation, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete mathematics, discrete problems by continuous ones.

### Modernization

In the 18th century, Leonhard Euler, Euler introduced the notion of function (mathematics), 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, Augustin Louis Cauchy, 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. Siméon Denis Poisson, Poisson, Joseph Liouville, Liouville, Joseph Fourier, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Karl Weierstrass, Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Bernhard Riemann, Riemann introduced his theory of integral, integration. The last third of the century saw the arithmetization of analysis by Karl Weierstrass, Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the (ε, δ)-definition of limit, "epsilon-delta" definition of limit of a function, limit. Then, mathematicians started worrying that they were assuming the existence of a Continuum (set theory), continuum of real numbers without proof. Richard Dedekind, 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 metric space, 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 integral, Riemann integration led to the study of the "size" of the set of Classification of discontinuities, discontinuities of real functions. Also, "pathological (mathematics), monsters" (nowhere continuous functions, continuous but Weierstrass function, nowhere differentiable functions, space-filling curves) began to be investigated. In this context, Camille Jordan, Jordan developed his theory of Jordan measure, measure, Georg Cantor, Cantor developed what is now called naive set theory, and René-Louis Baire, Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Henri Lebesgue, Lebesgue solved the problem of measure, and David Hilbert, Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Stefan Banach, Banach created functional analysis.

# Important concepts

## Metric spaces

In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a metric space is a Set (mathematics), set where a notion of distance (called a metric (mathematics), 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 $\left(M,d\right)$ where $M$ is a set and $d$ is a metric (mathematics), metric on $M$, i.e., a Function (mathematics), function :$d \colon M \times M \rightarrow \mathbb$ such that for any $x, y, z \in M$, the following holds: # $d\left(x,y\right) = 0$ if and only if $x = y$    (''identity of indiscernibles''), # $d\left(x,y\right) = d\left(y,x\right)$    (''symmetry''), and # $d\left(x,z\right) \le d\left(x,y\right) + d\left(y,z\right)$    (''triangle inequality''). By taking the third property and letting $z=x$, it can be shown that $d\left(x,y\right) \ge 0$     (''non-negative'').

## Sequences and limits

A sequence is an ordered list. Like a Set (mathematics), set, it contains Element (mathematics), 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 (mathematics), 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 (#Finite and infinite, 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 (''a''''n'') (with ''n'' running from 1 to infinity understood) the distance between ''a''''n'' and ''x'' approaches 0 as ''n'' → ∞, denoted :$\lim_ a_n = x.$

# 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 function (mathematics), functions and sequences, including Limit of a sequence, convergence and limit of a function, limits of sequences of real numbers, the
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

of the real numbers, and continuous function, continuity, smooth function, 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 function In mathematics, an analytic function is a function (mathematics), function that is locally given by a convergent series, convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are ...
s of complex variables (or, more generally, meromorphic functions). Because the separate real number, real and imaginary number, 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 space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definitions, topology, etc.) and the linear transformation, 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 function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations.

## Differential equations

A differential equation is a mathematics, mathematical equation for an unknown function (mathematics), function of one or several Variable (mathematics), variables that relates the values of the function itself and its derivatives of various Derivative#Higher derivatives, 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 system (mathematics), 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 of motion, 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 equations of motion, equation of motion) may be solved explicitly.

## Measure theory

A measure on a set (mathematics), 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 $n$-dimensional Euclidean space $\mathbb^n$. For instance, the Lebesgue measure of the Interval (mathematics), interval $\left\left[0, 1\right\right]$ in the real line, 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 Extended real number line, +∞ to (certain) subsets of a set $X$. 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 Sigma-algebra, $\sigma$-algebra. This means that countable union (set theory), unions, countable intersection (set theory), intersections and complement (set theory), 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 computation, 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 functional (mathematics), functionals, as opposed to ordinary
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

which deals with function (mathematics), functions. * Harmonic analysis deals with the representation of function (mathematics), functions or signals as the superposition principle, 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, ''p''-adic analysis, the study of analysis within the context of p-adic number, ''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 rigour#Mathematical rigour, rigorous treatment of infinitesimals and infinitely large numbers. * Computable analysis, the study of which parts of analysis can be carried out in a computability theory, 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).

# Applications

Techniques from analysis are also found in other areas such as:

## Physical sciences

The vast majority of classical mechanics, Theory of relativity, 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 Sound, 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

* Constructive analysis * History of calculus * Hypercomplex analysis * Non-classical analysis * Paraconsistent logic * Smooth infinitesimal analysis * Timeline of calculus and mathematical analysis

# References

*

(NB. 3 softcover volumes in slipcase. Original Russian title in March 1956: Математика, ее содержание, методы и значени

https://www.mathedu.ru/text/matematika_ee_soderzhanie_metody_i_znachenie_t2_1956

First English edition in 6 volumes by AMS in 1962/1963, revised English edition in 3 volumes by MIT Press in August 1964

2nd printing by MIT Press in April 1965. First MIT paperback edition in March 1969. Reprinted in one volume by Dover.) * * * * * * * * * (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992) *