Mathematical analysis
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/* See also */ [[Smooth infinitesimal analysis]]
'''Analysis''' has its beginnings in the rigorous formulation of [[calculus]]. It is the branch of [[mathematics]] most explicitly concerned with the notion of a [[limit (mathematics)|limit]], whether the [[limit of a sequence]] or the [[limit of a function]].<ref>(Whittaker and Watson, 1927, Chapter III)</ref> It also includes the theories of [[Derivative|differentiation]], [[Integral|integration]] and [[Measure (mathematics)|measure]], [[Series (mathematics)|infinite series]]<ref>Edwin Hewitt and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965</ref>, and [[analytic function]]s. These theories are often studied in the context of [[real number]]s, [[complex number]]s, and real and complex [[function (mathematics)|functions]]. However, they can also be defined and studied in any [[space#In mathematics|space]] of mathematical objects that is equipped with a definition of "nearness" (a [[topological space]]) or more specifically "distance" (a [[metric space]]).
==Motivation==
The motivation for studying mathematical analysis in the wider context of topological or metric spaces is twofold:
*First, the same basic techniques have proved applicable to a wider class of problems (e.g., the study of [[Functional Analysis|function spaces]]).
*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 series of complex exponentials or trigonometric functions. 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 dimensional space known as a [[Hilbert space]]. 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]]s, where this technique is known as separation of [[variables]].
== 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 of Elea|Zeno's]] [[Zeno's paradoxes|paradox of the dichotomy]].<ref name="Stillwell Infinite Series Early Results">{{cite book|last=Stillwell|authorlink=John Stillwell|title=|year=2004|chapter=Infinite Series|pages=170|quote=Infinite series were present in Greek mathematics, [...]There is no question that Zeno's paradox of the dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + 1/4 + 1/4^2 + 1/4^3 + ... = 4/3. Both these examples are special cases of the result we express as summation of a geometric series}}</ref> Later, [[Greek mathematics|Greek mathematicians]] such as [[Eudoxus of Cnidus|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.<ref>(Smith, 1958)</ref> In [[Indian mathematics|India]], the 12th century mathematician [[Bhaskara]] conceived of [[differential calculus]], and gave examples of the [[derivative]] and [[differential (infinitesimal)|differential]] coefficient, along with a statement of what is now known as [[Rolle's theorem]].
In the 14th century, the roots of mathematical analysis began with work done by [[Madhava of Sangamagrama]], regarded by some as the "founder of mathematical analysis",<ref>G. G. Joseph (1991). ''The crest of the peacock'', London.</ref> who developed [[series (mathematics)|infinite series]] expansions, like the [[power series]] and the [[Taylor series]], of functions such as [[sine]], [[cosine]], [[tangent (trigonometric function)|tangent]] and [[arctangent]]. Alongside his development of the Taylor series of the [[trigonometric function]]s 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]] further expanded his works, up to the 16th century.
In Europe, during the later half of the 17th century, [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] 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]], [[Ordinary differential equation|ordinary]] and [[partial differential equation]]s, [[Fourier analysis]], and [[generating function]]s. During this period, calculus techniques were applied to approximate [[discrete mathematics|discrete problems]] by continuous ones.
In the 18th century, [[Leonard Euler|Euler]] introduced the notion of [[function (mathematics)|mathematical function]].<ref name="function">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All| year = 1999| publisher =The Mathematical Association of America | pages = 17}}</ref> Real analysis began to emerge as an independent subject when [[Bernard Bolzano]] introduced the modern definition of continuity in 1816.<ref>*{{cite book|first=Roger |last=Cooke |authorlink=Roger Cooke |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |isbn=0471180823 |pages=379 |chapter=Beyond the Calculus |quote=Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781-1848).}}</ref> In the 19th century, [[Augustin Louis Cauchy|Cauchy]] helped to put calculus on a firm logical foundation by introducing the concept of the [[Cauchy sequence]]. He also started the formal theory of [[complex analysis]]. [[Simeon Poisson|Poisson]], [[Liouville]], [[Jean-Baptiste Joseph Fourier|Fourier]] and others studied partial differential equations and [[harmonic analysis]]. The contributions of these mathematicians and others, such as [[Karl Weierstrass|Weierstrass]], developed the modern notion of mathematical rigor, thus founding the field of mathematical analysis (at least in the modern sense).
In the middle of the century [[Bernhard Riemann|Riemann]] introduced his theory of [[integral|integration]]. The last third of the 19th century saw the arithmetization of analysis by [[Karl Weierstrass|Weierstrass]], who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of [[limit (mathematics)|limit]].
Then, mathematicians started worrying that they were assuming the existence of a [[continuum (mathematics)|continuum]] of [[real number]]s without proof. [[Julius Wilhelm Richard Dedekind|Dedekind]] then constructed the real numbers by [[Dedekind cut]]s, in which a mathematician creates irrational numbers that serve to fill the "gaps" between rational numbers, thereby creating a [[Complete space|complete]] set: the continuum of real numbers. Around that time, the attempts to refine the [[theorem]]s of [[Riemann integration]] led to the study of the "size" of the set of [[Classification of discontinuities|discontinuities]] of real functions.
Also, "[[pathological (mathematics)|monsters]]" ([[nowhere continuous]] functions, continuous but [[nowhere differentiable]] functions, [[space-filling curve]]s) began to be created. In this context, [[Camille Jordan|Jordan]] developed his theory of [[Jordan measure|measure]], [[Georg Cantor|Cantor]] developed what is now called [[naive set theory]], and [[Baire]] proved the [[Baire category theorem]]. In the early 20th century, calculus was formalized using [[axiomatic set theory]]. [[Henri Leon Lebesgue|Lebesgue]] solved the problem of measure, and [[David Hilbert|Hilbert]] introduced [[Hilbert space]]s to solve [[integral equation]]s. The idea of [[normed vector space]] was in the air, and in the 1920s [[Stefan Banach|Banach]] created [[functional analysis]].
== Subdivisions ==
Mathematical analysis includes the following subfields.
* [[Real analysis]], the [[rigour#Mathematical rigour|rigorous]] study of [[derivative]]s and [[integral]]s of functions of real variables. This includes the study of [[sequence]]s and their [[limit (mathematics)|limits]], [[series (mathematics)|series]], and [[measure (mathematics)|measure]]s.
* [[Functional analysis]]<ref>Carl L. Devito, "Functional Analysis", Academic Press, 1978</ref> studies spaces of functions and introduces concepts such as [[Banach space]]s and [[Hilbert space]]s.
* [[Harmonic analysis]] deals with [[Fourier series]] and their abstractions.
* [[Complex analysis]], the study of functions from the [[complex plane]] to the complex plane which are complex differentiable.
* [[Differential geometry and topology]], the application of calculus to abstract mathematical spaces that possess a complicated internal structure.
* [[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 number]]s and their functions and gives a [[rigour#Mathematical rigour|rigorous]] treatment of [[infinitesimal]]s and infinitely large numbers. It is normally classed as [[model theory]].
* [[Numerical analysis]], 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]]s is now shared with other fields such as [[dynamical system]]s, though the overlap with conventional analysis is large.
== See also ==
* [[Calculus]]
* [[Functional analysis]]
* [[Infinitesimal]]
* [[Method of exhaustion]]
* [[Non-classical analysis]]
* [[Non-standard analysis]]
* [[Smooth infinitesimal analysis]]
== Notes ==
{{reflist}}
==References==
* [[Walter Rudin]], Principles of Mathematical Analysis, McGraw-Hill Publishing Co.; 3Rev Ed edition (September 1, 1976), ISBN 978-0070856134.
*Apostol, Tom M., ''Mathematical Analysis'', 2nd ed. Addison-Wesley, 1974. ISBN 978-0201002881.
*Nikol'skii, S. M., [http://eom.springer.de/M/m062610.htm "Mathematical analysis"], in [http://eom.springer.de/default.htm ''Encyclopaedia of Mathematics''], Michiel Hazewinkel (editor), Springer-Verlag (2002). ISBN 1-4020-0609-8.
*Smith, David E., ''History of Mathematics'', Dover Publications, 1958. ISBN 0-486-20430-8.
*{{cite book
| first=John
| last=Stillwell
| authorlink=John Stillwell
| title=Mathematics and its History
| edition=Second Edition
| publisher=Springer Science + Business Media Inc.
| year=2004
| isbn=0387953361
}}
*[[E. T. Whittaker|Whittaker, E. T.]] and [[G. N. Watson|Watson, G. N.]], ''A Course of Modern Analysis'', fourth edition, Cambridge University Press, 1927. ISBN 0521588073.
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