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Analysis is the branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
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 ...

differentiation
,
integration
integration
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 ...

complex
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 infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
; 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 Gre ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
) or specific distances between objects (a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
).


History


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 In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, ...

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 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#REDIRECT 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 Eu ...

Archimedes
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 Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
to compute the area and volume of regions and solids. The explicit use of
infinitesimals In mathematics, an infinitesimal or infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimu ...
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 Greek ...
'', 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 perio ...
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, έχνη ''tiké échne', 'art' or 'cr ...
and
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ...
series as early as the 4th century B.C. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 B.C. In
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
, particular instances of arithmetic series have been found to implicitly occur in Vedic Literature as early as 2000 B.C.


Medieval

Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the East Asian cultural sphere ...
established a method that would later be called
Cavalieri's principle In 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 t ...
to find the volume of a
sphere A sphere (from Greek#REDIRECT 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 population is appr ...

sphere
in the 5th century. In the 12th century, the Indian mathematician
Bhāskara II Bhāskara (c. 1114–1185) also known as Bhāskarācārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian people, Indian Indian mathematicians, mathematician and astronomer. He was born in Bij ...
gave examples of
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
s and used what is now known as
Rolle's theorem In 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. ...

Rolle's theorem
. In the 14th century,
Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan Nampūtiri known as Mādhava of Sangamagrāma () was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes t ...
developed
infinite series 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). I ...
expansions, now called
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, of functions such as
sine 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). ...

sine
,
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
,
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
and
arctangent In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Alongside his development of Taylor series of
trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
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 Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the '' Parlement'' of Toulouse Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

Fermat
and developed
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, which is the precursor to modern calculus. Fermat's method of
adequality Adequality is a technique developed by Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French people, French mathematician who is given credit for early developments that led to infinitesimal ...
allowed him to determine the maxima and minima of functions and the tangents of curves. Descartes' publication of
La Géométrie ''La Géométrie'' was published Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term refers to the distribution of printed works ...
in 1637, which introduced the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
, is considered to be the establishment of mathematical analysis. It would be a few decades later that
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

Newton
and
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
independently developed
infinitesimal 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 grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
,
ordinary Ordinary or The Ordinary often refer to: Music * Ordinary (EP), ''Ordinary'' (EP) (2015), by South Korean group Beast * Ordinary (Every Little Thing album), ''Ordinary'' (Every Little Thing album) (2011) * Ordinary (Two Door Cinema Club song), "O ...
and
partial differential equation 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). I ...
s,
Fourier analysis 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 ...
, and
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...
s. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.


Modernization

In the 18th century,
Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Euler
introduced the notion of
mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. Real analysis began to emerge as an independent subject when
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost a ...

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 Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

Cauchy
began to put calculus on a firm logical foundation by rejecting the principle of the
generality of algebraIn the history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of ...
widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and
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 exist in the standard real number system, but do exist in many other number systems, such a ...
s. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and started the formal theory of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
. Poisson,
Liouville Joseph Liouville FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently dist ...

Liouville
, and others studied partial differential equations and
harmonic analysis Harmonic analysis is a 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 (mathe ...
. The contributions of these mathematicians and others, such as
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
introduced his theory of . The last third of the century saw the arithmetization of analysis by
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

Weierstrass
, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

limit
. Then, mathematicians started worrying that they were assuming the existence of a
continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variatio ...
of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s without proof. Dedekind then constructed the real numbers by
Dedekind cut 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). ...
s, 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 Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
in terms of decimal expansions. Around that time, the attempts to refine the
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s of
Riemann integration The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
led to the study of the "size" of the set of discontinuities of real functions. Also, "
monsters A monster is a type of fictional creature found in Horror fiction, horror, fantasy, science fiction, folklore, mythology and religion. Monsters are very often depicted as dangerous and aggressive with a strange, grotesque appearance that causes ...
" ( nowhere continuous functions, continuous but ,
space-filling curve In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
s) began to be investigated. In this context,
Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ') is a country in Western Asia Western Asia, West Asia, or Southwest Asia, is the westernmost subregion A subregion is a part of a larger region In ge ...
developed his theory of measure,
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer Prayer is an invocation An invocation (from the Latin verb ''invocare'' "to call on, invoke, to give") may take the form of: * Supplication, prayer ...
developed what is now called
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
, and Baire proved the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set t ...
. In the early 20th century, calculus was formalized using an axiomatic
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
.
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a France, French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area betwe ...
solved the problem of measure, and
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
introduced
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s to solve
integral equation 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 ...
s. The idea of
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
was in the air, and in the 1920s created
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
.


Important concepts


Metric spaces

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a metric space is a set where a notion of
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

distance
(called a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
,
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, other
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, and the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s. Examples of analysis without a metric include
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
(which describes size rather than distance) and
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
(which studies
topological vector 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 ...
s that need not have any sense of distance). Formally, a metric space is an
ordered pair 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 h ...

ordered pair
(M,d) where M is a set and d is a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
on M, i.e., a
function 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 ...
:d \colon M \times M \rightarrow \mathbb such that for any x, y, z \in M, the following holds: # d(x,y) = 0
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
x = y    (''
identity of indiscernibles The identity of indiscernibles is an ontology, ontological principle that states that there cannot be separate object (philosophy), objects or wikt:entity, entities that have all their property (philosophy), properties in common. That is, entities ' ...
''), # d(x,y) = d(y,x)    (''symmetry''), and # d(x,z) \le d(x,y) + d(y,z)    (''
triangle inequality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

triangle inequality
''). By taking the third property and letting z=x, it can be shown that d(x,y) \ge 0     (''non-negative'').


Sequences and limits

A sequence is an ordered list. Like a set, it contains
members Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ...
(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 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 ...
whose domain is a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
totally ordered 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). I ...
set, such as the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

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 (''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 number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and real-valued functions of a real variable. In particular, it deals with the analytic properties of real
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 ...
and
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
s, including
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
and
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 ...

limits
of
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
s of real numbers, the
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
of the real numbers, and continuity,
smoothness is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...

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 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). I ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s 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 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 one or several Variable (mathematics), variables that relates the values of the function itself and its
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
s 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 Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, 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[0, 1\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.


Vector analysis


Tensor analysis


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 infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
which deals with
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 ...
. * Harmonic analysis deals with the representation of
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 ...
or signals as the superposition principle, superposition of basic waves. * Geometric analysis involves the use of geometrical methods in the study of
partial differential equation 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). I ...
s 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
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 exist in the standard real number system, but do exist in many other number systems, such a ...
s 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


See also

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


References


Further reading

*

(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) *


External links


Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis

Basic Analysis: Introduction to Real Analysis
by Jiri Lebl (Creative Commons, Creative Commons BY-NC-SA)
Mathematical Analysis-Encyclopædia Britannica


{{Authority control Mathematical analysis,