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Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
. It is the branch of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 most explicitly concerned with the notion of a limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
, whether the limit of a sequence
Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit....
 or the limit of a function
Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and mathematical analysis concerning the behavior of that Function near a particular independent variable....
. It also includes the theories of differentiation
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
, integration
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 and measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
, infinite series
Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
, and analytic function
Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions....
s. These theories are often studied in the context of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, and real and complex functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
. However, they can also be defined and studied in any space
Space

Space is the boundless, three-dimensional extent in which Physical body and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physics usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime....
 of mathematical objects that has a definition of nearness (a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
) or, more specifically, distance (a metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
).

motivation for studying mathematical analysis in the wider context of toplogical
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 or metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
s is two-fold:



y results in analysis were implicitly present in the early days of ancient Greek mathematics.






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Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
. It is the branch of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 most explicitly concerned with the notion of a limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
, whether the limit of a sequence
Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit....
 or the limit of a function
Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and mathematical analysis concerning the behavior of that Function near a particular independent variable....
. It also includes the theories of differentiation
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
, integration
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 and measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
, infinite series
Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
, and analytic function
Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions....
s. These theories are often studied in the context of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, and real and complex functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
. However, they can also be defined and studied in any space
Space

Space is the boundless, three-dimensional extent in which Physical body and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physics usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime....
 of mathematical objects that has a definition of nearness (a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
) or, more specifically, distance (a metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
).

Motivation

The motivation for studying mathematical analysis in the wider context of toplogical
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 or metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
s is two-fold:
  • First, the same basic techniques have proved applicable to a wider class of problems (e.g., the study of function spaces
    Functional analysis

    Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
    ).


  • Second, and just as importantly, a greater understanding of analysis in more abstract spaces frequently proves to be directly applicable to classical problems. For example, in Fourier analysis, functions are expressed in terms of certain infinite sum of trigonometric function
    Trigonometric function

    In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
    s. Thus Fourier analysis might be used to decompose a (possibly very complicated) sound into a unique combination of pure tones of various pitches. The "weights", or coefficients, of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimension
    Dimension

    In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
    al space known as a Hilbert space
    Hilbert space

    The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
    . Study of functions defined in this more general setting thus provides a convenient method of deriving results about the way functions vary in space as well as time or, in more mathematical terms, partial differential equation
    Partial differential equation

    In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
    s, where this technique is known as separation of variables
    Separation of variables

    In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
    .


History

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
Zeno of Elea

Zeno of Velia was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic....
 paradox of the dichotomy
Zeno's paradoxes

Zeno's paradoxes are a set of problems generally thought to have been devised by Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion....
. Later, Greek mathematicians
Greek mathematics

Greek mathematics, as that term is used in this article, is the mathematics written in Greek language, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean....
 such as Eudoxus
Eudoxus of Cnidus

Eudoxus of Cnidus was a Ancient Greece astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy....
 and Archimedes
Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
 made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion
Method of exhaustion

The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas Convergence to the area of the containing shape....
 to compute the area and volume of regions and solids. In India
Indian mathematics

Indian mathematics—which here is the mathematics that emerged in South Asia from ancient times until the end of the 18th century—had its beginnings in the Bronze Age Indus Valley civilization and the Iron Age Vedic culture ....
, the 12th century mathematician Bhaskara II conceived of differential calculus
Differential calculus

Differential calculus, a field in mathematics, is the study of how function s change when their inputs change. The primary object of study in differential calculus is the derivative....
, and gave examples of the derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
 and differential coefficient, along with a statement of what is now known as Rolle's theorem
Rolle's theorem

In calculus, a branch of mathematics, Rolle's theorem essentially states that a differentiable function , which attains equal values at two points, must have a stationary point somewhere between them where the slope is zero....
.

In the 14th century, the roots of mathematical analysis began with work done by Madhava of Sangamagrama
Madhava of Sangamagrama

Madhava of Sangamagrama was a prominent Indian mathematics-Indian astronomy from the town of Irinjalakkuda, near Cochin, Kerala, India, which was at the time known as Sangamagrama ....
, regarded by some as the "founder of mathematical analysis", who developed infinite series
Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
 expansions, like the power series
Power series

In mathematics, a power series is an infinite series of the formwhere an represents the coefficient of the nth term, c is a constant, and x varies around c ....
 and the Taylor series
Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
, 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.

In Europe, during the later half of the 17th century, Newton
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
 and Leibniz
Gottfried Leibniz

Gottfried Wilhelm Leibniz was a Germany polymath who wrote primarily in Latin and French language.He occupies an equally grand place in both the history of philosophy and the history of mathematics....
 independently developed calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations
Calculus of variations

Calculus of variations is a field of mathematics that deals with functional , as opposed to ordinary calculus which deals with function . Such functionals can for example be formed as integrals involving an unknown function and its derivatives....
, ordinary
Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....
 and partial differential equation
Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s, Fourier analysis, and generating function
Generating function

In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers....
s. During this period, calculus techniques were applied to approximate discrete problems
Discrete mathematics

Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense that its objects can assume only distinct, separate values, rather than a values on a continuum ....
 by continuous ones.

In the 18th century, Euler
Leonhard Euler

Leonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany.Euler made important discoveries in fields as diverse as calculus and graph theory....
 introduced the notion of mathematical function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
. Real analysis began to emerge as an independent subject when Bernard Bolzano
Bernard Bolzano

Bernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, theology, philosopher, logician and antimilitarism of German language mother tongue....
 introduced the modern definition of continuity in 1816. In the 19th century, Cauchy helped to put calculus on a firm logical foundation by introducing the concept of the Cauchy sequence
Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses....
. He also started the formal theory of complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
. Poisson
Siméon Denis Poisson

Sim?on-Denis Poisson , was a France mathematician, geometer, and physicist. The name is in French language....
, Liouville
Joseph Liouville

Joseph Liouville was a France mathematician....
, Fourier
Joseph Fourier

Jean Baptiste Joseph Fourier was a France mathematician and physicist best known for initiating the investigation of Fourier series and their application to problems of heat flow....
 and others studied partial differential equations and harmonic analysis
Harmonic analysis

Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms....
. The contributions of these mathematicians and others, such as Weierstrass
Karl Weierstrass

Karl Theodor Wilhelm Weierstrass was a Germany mathematics who is often cited as the "father of modern mathematical analysis"....
, developed the modern notion of mathematical rigor, thus founding the field of mathematical analysis (at least in the modern sense).

In the middle of the century Riemann
Bernhard Riemann

Georg Friedrich Bernhard Riemann was a Germany mathematics who made important contributions to mathematical analysis and differential geometry, some of them paving the way for the later development of general relativity....
 introduced his theory of integration
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
. The last third of the 19th century saw the arithmetization of analysis by Weierstrass
Karl Weierstrass

Karl Theodor Wilhelm Weierstrass was a Germany mathematics who is often cited as the "father of modern mathematical analysis"....
, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of limit
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
. Then, mathematicians started worrying that they were assuming the existence of a continuum
Continuum (mathematics)

In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum....
 of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s without proof. Dedekind
Richard Dedekind

Julius Wilhelm Richard Dedekind was a Germany mathematics who did important work in abstract algebra, algebraic number theory and the foundations of the real numbers....
 then constructed the real numbers by Dedekind cut
Dedekind cut

In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of a set of it into two non-empty parts, , such that A is closed downwards and B is closed upwards, and A contains no greatest element....
s, in which a mathematician creates irrational numbers that serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers. Around that time, the attempts to refine the theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
s of Riemann integration
Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an Interval ....
 led to the study of the "size" of the set of discontinuities
Classification of discontinuities

Continuous functions are of utmost importance in mathematics and applications. However, not all function are continuous. If a function is not continuous at a point in its domain , one says that it has a discontinuity there....
 of real functions.

Also, "monsters
Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive.Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological....
" (nowhere continuous functions, continuous but nowhere differentiable functions
Weierstrass function

In mathematics, the Weierstrass function is a pathological example of a real line-valued function on the real line. The function has the property that it is continuous function everywhere but differentiable nowhere....
, space-filling curve
Space-filling curve

In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square . Because Giuseppe Peano was the first to discover one, space-filling curves in the Plane are commonly called Peano curves....
s) began to be created. In this context, Jordan
Camille Jordan

Marie Ennemond Camille Jordan was a France mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse....
 developed his theory of measure
Jordan measure

In mathematics, the Jordan measure is an extension of the notion of size to shapes more complicated than, for example, a triangle, Disk , or parallelipiped....
, Cantor
Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a Germany mathematician, born in Russia. He is best known as the creator of set theory, which has become a foundations of mathematics in mathematics....
 developed what is now called naive set theory
Naive set theory

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics....
, and Baire
René-Louis Baire

Ren?-Louis Baire , was a France mathematician. He was born in Paris, France and died in Chamb?ry, France.Dogged by ill health, and spending time alternating between low level teaching in lyc?es and work in universities, he was only able to make contributions to mathematics in limited spells....
 proved the Baire category theorem
Baire category theorem

The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
. In the early 20th century, calculus was formalized using an axiomatic set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
. Lebesgue
Henri Lebesgue

Henri L?on Lebesgue was a France mathematician, most famous for his theory of integral. Lebesgue's integration theory was originally published in his dissertation, A summary of Henri Lebesgue's dissertation , at the University of Nancy in 1902....
 solved the problem of measure, and Hilbert
David Hilbert

David Hilbert was a Germany mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries....
 introduced Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
s to solve integral equation
Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential equation and integral equations, and some problems may be formulated either way....
s. The idea of normed vector space
Normed vector space

In mathematics, with 2- or 3-dimensional Vector s with real number-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any Vector space Rn....
 was in the air, and in the 1920s Banach
Stefan Banach

Stefan Banach was a Polish mathematician who worked in Second Polish Republic and in Soviet Ukraine.A self-taught mathematics Child prodigy, Banach was the founder of modern functional analysis and a founder of the Lw?w School of Mathematics....
 created functional analysis
Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
.

Subdivisions

Mathematical analysis includes the following subfields.
  • Real analysis
    Real analysis

    Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the Set of real numbers. In particular, it deals with the analytic function properties of real function and sequences, including convergence and limit s of sequences of real numbers, the calculus of the real numbers, and continu...
    , the rigorous
    Rigour

    Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism....
     study of derivative
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
    s and integral
    Integral

    Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
    s of functions of real variables. This includes the study of sequence
    Sequence

    In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
    s and their limits
    Limit (mathematics)

    In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
    , series
    Series (mathematics)

    In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
    , and measure
    Measure (mathematics)

    In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
    s.
  • Functional analysis
    Functional analysis

    Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
     studies spaces of functions and introduces concepts such as Banach space
    Banach space

    In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them....
    s and Hilbert space
    Hilbert space

    The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
    s.
  • Harmonic analysis
    Harmonic analysis

    Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms....
     deals with Fourier series
    Fourier series

    In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sine wave . The study of Fourier series is a branch of Fourier analysis....
     and their abstractions.
  • Complex analysis
    Complex analysis

    Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
    , the study of functions from the complex plane
    Complex plane

    In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
     to the complex plane which are complex differentiable.
  • General topology
    General topology

    In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them....
    , the study of topological tranformations of an arbitrary geometric figure.
  • Differential geometry, the application of calculus to abstract mathematical spaces that possess a complicated internal structure.
  • p-adic analysis
    P-adic analysis

    In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.The theory of complex-valued numerical functions on the p-adic numbers is just part of the theory of locally compact groups....
    , the study of analysis within the context of p-adic numbers
    P-adic number

    In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real number and complex number systems....
    , which differs in some interesting and surprising ways from its real and complex counterparts.
  • Non-standard analysis
    Non-standard analysis

    Non-standard analysis is a branch of mathematics that formulates mathematical analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson....
    , which investigates the hyperreal number
    Hyperreal number

    The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Isaac Newton and Gottfried Leibniz....
    s and their functions and gives a rigorous
    Rigour

    Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism....
     treatment of infinitesimal
    Infinitesimal

    Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. For everyday life, an infinitesimal object is an object which is smaller than any possible measure....
    s and infinitely large numbers. It is normally classed as model theory
    Model theory

    In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
    .
  • Numerical analysis
    Numerical analysis

    Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
    , the study of algorithms for approximating the problems of continuous mathematics.


Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equation
Differential equation

A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....
s is now shared with other fields such as dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
s, though the overlap with conventional analysis is large.

See also

  • Method of exhaustion
    Method of exhaustion

    The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas Convergence to the area of the containing shape....
  • Non-classical analysis
    Non-classical analysis

    In mathematics, non-classical analysis is any system of analysis, other than classical real analysis, and complex, vector, tensor, etc., analysis based upon it....
  • Smooth infinitesimal analysis
    Smooth infinitesimal analysis

    Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W....