mathematical analysis

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Analysis is the branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, an ...
; however, it can be applied to any
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics Classical physics is a group of physics Physics is the natural science that studies matte ...
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 Deductive reason ...
s that has a definition of nearness (a topological space) or specific distances between objects (a
metric space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
).

# History

## Ancient

Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' '' The Method of Mechanical Theorems'', a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric 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 The Indian subcontinent is a physiographical region in Southern Asia. It is situated on the Indian Plate, projecting southwards into the Indian Ocean The Indian Ocean is the t ...
, 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 In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they a ...
to find the volume of a sphere in the 5th century. In the 12th century, the 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 infinite series expansions, now called Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of Taylor series of
trigonometric functions In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, 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's 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 Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations,
Fourier analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

### Modernization

In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in ''x'' to correspond to an infinitesimal change in ''y''. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration, and made significant advances in complex analysis. Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of
real number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. Also, various pathological objects, (such as nowhere continuous functions, continuous but nowhere differentiable functions, and
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, ...
s), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic
set theory Set theory is the branch of mathematical logic Mathematical logic is the study of formal logic within mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...
. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved to be a big improvement over Riemann's. Hilbert introduced
Hilbert space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

# Important concepts

## Metric spaces

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a metric space is a set where a notion of distance (called a
metric Metric or metrical may refer to: * Metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these s ...
) 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 Euclidean space is the fundamental space of geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structure ...
, other
vector space In mathematics and physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is th ...
s, and the
integer An integer is the number zero (), a positive natural number In mathematics, the natural numbers are those number A number is a mathematical object used to count, measure, and label. The original examples are the natural number ...
s. 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 Metric or metrical may refer to: * Metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these s ...
on $M$, i.e., a function :$d \colon M \times M \rightarrow \mathbb$ such that for any $x, y, z \in M$, the following holds: # $d\left(x,y\right) \geq 0$, with equality 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, it contains members (also called ''elements'', or ''terms''). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a
countable In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
totally ordered set, such as the
natural numbers In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
. 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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s, including convergence and limits of
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

## Complex analysis

Complex analysis (traditionally known as the theory of functions of a complex variable) is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number In mathematics, the natura ...
, applied mathematics; as well as in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, including hydrodynamics, thermodynamics,
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics Mathematics is an area of knowledge that includes the topics o ...
, electrical engineering, and particularly,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics Quantum mechanics is a fundamental theory in physics Physics is th ...
. Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of
vector space In mathematics and physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is th ...
s endowed with some kind of limit-related structure (e.g. inner product, norm,
topology In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time Time is the contin ...
as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

## Harmonic analysis

Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic
wave In physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s. This includes the study of the notions of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ' ...
and
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time Time is the contin ...
s (
Fourier analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
), and of their generalizations. Harmonic analysis has applications in areas as diverse as
music theory Music theory is the study of the practices and possibilities of music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number In mathematics, the natura ...
, representation theory,
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound In physics Physics is the natural science that studies matter, its fundamental cons ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and f ...
, tidal analysis, and neuroscience.

## Differential equations

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics ana ...
,
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process heredita ...
, and other disciplines. Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft A spacecraft is a vehicle or machine designed to fly i ...
, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

## Measure theory

A measure on a set is a systematic way to assign a number to each suitable
subset In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
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 In measure theory, a branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These ...
on a
Euclidean space Euclidean space is the fundamental space of geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structure ...
, which assigns the conventional length,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ...
, and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics Mathematics ...
to suitable subsets of the $n$-dimensional Euclidean space $\mathbb^n$. For instance, the Lebesgue measure of the interval
real numbers In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set $X$. It must assign 0 to the
empty set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.

## Numerical analysis

Numerical analysis is the study of
algorithm In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represen ...
s that use numerical
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin Latin (, or , ) is a classical language belonging to the Italic lan ...
(as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

## Vector analysis

Vector analysis is a branch of mathematical analysis dealing with values which have both magnitude and direction. Some examples of vectors include velocity, force, and displacement. Vectors are commonly associated with scalars, values which describe magnitude.

## Scalar analysis

Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.

# Other topics

* Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. * Harmonic analysis deals with the representation of functions or signals as the superposition of basic
wave In physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s. * Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry. * Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions. * ''p''-adic analysis, the study of analysis within the context of ''p''-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts. * Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. * Computable analysis, the study of which parts of analysis can be carried out in a computable manner. *
Stochastic calculus Stochastic calculus is a branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. T ...
– 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). * Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory. * Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences. * Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory. * Smooth infinitesimal analysis, which is developed in a smooth topos.

# Applications

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

## Physical sciences

The vast majority of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft A spacecraft is a vehicle or machine designed to fly i ...
, relativity, and
quantum mechanics Quantum mechanics is a fundamental theory in physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and f ...
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 Quantum mechanics is a fundamental theory in physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and f ...
.

## Signal processing

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.

## Other areas of mathematics

Techniques from analysis are used in many areas of mathematics, including: * Analytic number theory * Analytic combinatorics * Continuous probability * Differential entropy in information theory * Differential games * Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally. * Differentiable manifolds * Differential topology * Partial differential equations

# Famous Textbooks

* Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau * Introductory Real Analysis, by Andrey Kolmogorov, Sergei Fomin * Differential and Integral Calculus (3 volumes), by Grigorii Fichtenholz * The Fundamentals of Mathematical Analysis (2 volumes), by Grigorii Fichtenholz * A Course Of Mathematical Analysis (2 volumes), by Sergey Nikolsky * Mathematical Analysis (2 volumes), by Vladimir Zorich * A Course of Higher Mathematics (5 volumes, 6 parts), by Vladimir Smirnov * Differential And Integral Calculus, by Nikolai Piskunov * A Course of Mathematical Analysis, by Aleksandr Khinchin * Mathematical Analysis: A Special Course, by Georgiy Shilov * Theory of Functions of a Real Variable (2 volumes), by Isidor Natanson * Problems in Mathematical Analysis, by Boris Demidovich * Problems and Theorems in Analysis (2 volumes), by George Polya, Gabor Szegö * Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol * Principles of Mathematical Analysis, by Walter Rudin * Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein * Complex Analysis, by Elias Stein * Functional Analysis: Introduction to Further Topics in Analysis, by Elias Stein * Analysis (2 volumes), by Terence Tao * Analysis (3 volumes), by Herbert Amann, Joachim Escher * Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov * Real and Functional Analysis, by Serge Lang

* Constructive analysis * History of calculus * Hypercomplex analysis * Multivariable calculus * 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) *