Editing Mathematical analysis (section)

== History ==
[[Image:Archimedes pi.svg|thumb|right|300px|[[Archimedes]] used the [[method of exhaustion]] to compute the [[area]] inside a circle by finding the area of [[regular polygon]]s with more and more sides. This was an early but informal example of a [[limit (mathematics)|limit]], one of the most basic concepts in mathematical analysis.]]

===Ancient===
Mathematical analysis formally developed in the 17th century during the [[Scientific Revolution]],<ref name=analysis>{{cite book|last=Jahnke|first=Hans Niels|title=A History of Analysis|series=History of Mathematics |url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7|date=2003|volume=24 |publisher=[[American Mathematical Society]]|isbn=978-0821826232|page=7|access-date=2015-11-15|archive-date=2016-05-17|archive-url=https://web.archive.org/web/20160517180439/https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7|url-status=live|doi=10.1090/hmath/024}}</ref> but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of [[Greek mathematics|ancient Greek mathematics]]. For instance, an [[geometric series|infinite geometric sum]] is implicit in [[Zeno of Elea|Zeno's]] [[Zeno's paradoxes#Dichotomy paradox|paradox of the dichotomy]].<ref name="Stillwell_2004"/> (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) 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 name="Smith_1958"/> The explicit use of [[infinitesimals]] appears in Archimedes' ''[[The Method of Mechanical Theorems]]'', a work rediscovered in the 20th century.<ref>{{cite book|last=Pinto|first=J. Sousa|title=Infinitesimal Methods of Mathematical Analysis|url=https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7|date=2004|publisher=Horwood Publishing|isbn=978-1898563990|page=8|access-date=2015-11-15|archive-date=2016-06-11|archive-url=https://web.archive.org/web/20160611045431/https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7|url-status=live}}</ref> In Asia, the [[Chinese mathematics|Chinese mathematician]] [[Liu Hui]] used the method of exhaustion in the 3rd century CE to find the area of a circle.<ref>{{cite book|series=Chinese studies in the history and philosophy of science and technology|volume=130|title=A comparison of Archimedes' and Liu Hui's studies of circles|first1=Liu|last1=Dun|first2=Dainian|last2=Fan|first3=Robert Sonné|last3=Cohen|publisher=Springer|date=1966|isbn=978-0-7923-3463-7|page=279|url=https://books.google.com/books?id=jaQH6_8Ju-MC|access-date=2015-11-15|archive-date=2016-06-17|archive-url=https://web.archive.org/web/20160617055211/https://books.google.com/books?id=jaQH6_8Ju-MC|url-status=live}}, [https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 Chapter, p. 279] {{Webarchive|url=https://web.archive.org/web/20160526221958/https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 |date=2016-05-26 }}</ref> From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the [[arithmetic series|arithmetic]] and [[geometric series|geometric]] series as early as the 4th century BCE.<ref>{{cite journal | title = On the Use of Series in Hindu Mathematics | author = Singh, A. N. | journal  =    Osiris | volume = 1  |date = 1936  | pages = 606–628 | doi = 10.1086/368443 | jstor = 301627 | s2cid = 144760421 | url = https://www.jstor.org/stable/301627}}</ref>
[[Bhadrabahu|Ācārya Bhadrabāhu]] uses the sum of a geometric series in his Kalpasūtra in {{BCE|433}}.<ref>{{cite journal | title = Summation of Convergent Geometric Series and the concept of approachable Sunya | author = K. B. Basant, Satyananda Panda | journal = Indian Journal of History of Science | volume = 48  |date = 2013  | pages = 291–313 | url = https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol48_2_7_KBBasant.pdf}}</ref>

===Medieval===
[[Zu Chongzhi]] established a method that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]] in the 5th century.<ref>{{cite book|title=Calculus: Early Transcendentals|edition=3|first1=Dennis G.|last1=Zill|first2=Scott|last2=Wright|first3=Warren S.|last3=Wright|publisher=Jones & Bartlett Learning|date=2009|isbn=978-0763759957|page=xxvii|url=https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27|access-date=2015-11-15|archive-date=2019-04-21|archive-url=https://web.archive.org/web/20190421114230/https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27|url-status=live}}</ref> In the 12th century, the [[Indian mathematics|Indian mathematician]] [[Bhāskara II]] used infinitesimal and used what is now known as [[Rolle's theorem]].<ref>{{citation|title=The positive sciences of the ancient Hindus|journal=Nature|volume=97|issue=2426|page=177|first=Sir Brajendranath|last=Seal|date=1915|bibcode=1916Natur..97..177.|doi=10.1038/097177a0|hdl=2027/mdp.39015004845684|s2cid=3958488|hdl-access=free}}</ref>

In the 14th century, [[Madhava of Sangamagrama]] developed [[series (mathematics)|infinite series]] expansions, now called [[Taylor series]], of functions such as [[Trigonometric functions|sine]], [[Trigonometric functions|cosine]], [[trigonometric functions|tangent]] and [[Inverse trigonometric functions|arctangent]].<ref name=rajag78>
{{cite journal
 | title =       On an untapped source of medieval Keralese Mathematics
 | first1=      C. T.
 | last1=      Rajagopal
 | first2 =      M. S.
 | last2=      Rangachari
 | journal  =    Archive for History of Exact Sciences
 | volume =      18 | number=2
 |date=June 1978
 | pages =       89–102 | doi = 10.1007/BF00348142
| s2cid= 51861422
 }}</ref> Alongside his development of Taylor series of [[trigonometric functions]], he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series.  His followers at the [[Kerala School of Astronomy and Mathematics]] further expanded his works, up to the 16th century.

===Modern===
====Foundations====
The modern foundations of mathematical analysis were established in 17th century Europe.<ref name=analysis/> 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.<ref name=Pellegrino>{{cite web | last = Pellegrino | first = Dana | title = Pierre de Fermat | url = http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | access-date = 2008-02-24 | archive-date = 2008-10-12 | archive-url = https://web.archive.org/web/20081012024028/http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | url-status = live }}</ref> 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 [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] independently developed [[infinitesimal calculus]], which grew, with the stimulus of applied work that continued through the 18th&nbsp;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.

====Modernization====
In the 18th century, [[Leonhard Euler|Euler]] introduced the notion of a [[function (mathematics)|mathematical function]].<ref name="function">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All| url = https://archive.org/details/eulermasterofusa0000dunh| url-access = registration| date = 1999| publisher =The Mathematical Association of America | page= [https://archive.org/details/eulermasterofusa0000dunh/page/17 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 |author-link=Roger Cooke (mathematician) |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |date=1997 |isbn=978-0471180821 |page=[https://archive.org/details/historyofmathema0000cook/page/379 379] |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) |chapter-url=https://archive.org/details/historyofmathema0000cook/page/379 }}</ref> but Bolzano's work did not become widely known until the 1870s. In 1821, [[Augustin Louis Cauchy|Cauchy]] began to put calculus on a firm logical foundation by rejecting the principle of the [[generality of algebra]] widely used in earlier work, particularly by Euler.  Instead, Cauchy formulated calculus in terms of geometric ideas and [[infinitesimal]]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]], and started the formal theory of [[complex analysis]]. [[Siméon Denis Poisson|Poisson]], [[Joseph Liouville|Liouville]], [[Joseph Fourier|Fourier]] and others studied partial differential equations and [[harmonic analysis]].  The contributions of these mathematicians and others, such as [[Karl Weierstrass|Weierstrass]], developed the [[(ε, δ)-definition of limit]] approach, thus founding the modern field of mathematical analysis. Around the same time, [[Bernhard Riemann|Riemann]] introduced his theory of [[integral|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 (set theory)|continuum]] of [[real number]]s without proof. [[Richard Dedekind|Dedekind]] then constructed the real numbers by [[Dedekind cut]]s, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a [[complete metric space|complete]] set: the continuum of real numbers, which had already been developed by [[Simon Stevin]] in terms of [[decimal expansion]]s. Around that time, the attempts to refine the [[theorem]]s of [[Riemann integral|Riemann integration]] led to the study of the "size" of the set of [[Classification of discontinuities|discontinuities]] of real functions.

Also, various [[pathological (mathematics)|pathological objects]], (such as [[nowhere continuous function]]s, continuous but [[Weierstrass function|nowhere differentiable functions]], and [[space-filling curve]]s), commonly known as "monsters", began to be investigated. In this context, [[Camille Jordan|Jordan]] developed his theory of [[Jordan measure|measure]], [[Georg Cantor|Cantor]] developed what is now called [[naive set theory]], and [[René-Louis Baire|Baire]] proved the [[Baire category theorem]]. In the early 20th century, calculus was formalized using an axiomatic [[set theory]]. [[Henri Lebesgue|Lebesgue]] 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. [[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]].