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{{Fourier transforms}}
[[File:Fourier Series.svg|thumb|right|180px|The first four partial sums of the Fourier series for a [[square wave]]]]
In [[mathematics]], a '''Fourier series''' decomposes [[periodic function]]s or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely [[sine wave|sines and cosines]] (or [[complex exponential]]s). The study of Fourier series is a branch of [[Fourier analysis]].
The Fourier series is named in honour of [[Jean-Baptiste Joseph Fourier]] (1768–1830), who made important contributions to the study of [[trigonometric series]], after preliminary investigations by [[Leonhard Euler]], [[Jean le Rond d'Alembert]], and [[Daniel Bernoulli]].<ref group="nb">These three did some [[wave equation#Notes|important early work on the wave equation]], especially D'Alembert. Euler's work in this area was mostly [[Euler-Bernoulli beam equation|comtemporaneous/ in collaboration with Bernoulli]], although the latter made some independent contributions to the theory of waves and vibrations ([http://books.google.co.uk/books?id=olMpStYOlnoC&pg=PA214&lpg=PA214&dq=bernoulli+solution+wave+equation&source=bl&ots=h8eN69CWRm&sig=lRq2-8FZvcXIjToXQI4k6AVfRqA&hl=en&sa=X&ei=RqOhUIHOIOa00QWZuIHgCw&ved=0CCEQ6AEwATg8#v=onepage&q=bernoulli%20solution%20wave%20equation&f=false see here, pg.s 209 & 210, ]).</ref> Fourier introduced the series for the purpose of solving the [[heat equation]] in a metal plate, publishing his initial results in his 1807 ''[[Mémoire sur la propagation de la chaleur dans les corps solides]]'' (''Treatise on the propagation of heat in solid bodies''), and publishing his ''Théorie analytique de la chaleur'' in 1822. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on [[Deferent and epicycle|deferents and epicycles]].
The heat equation is a [[partial differential equation]]. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a [[sine]] or [[cosine]] wave. These simple solutions are now sometimes called [[Eigenvalue, eigenvector and eigenspace|eigensolutions]]. Fourier's idea was to model a complicated heat source as a superposition (or [[linear combination]]) of simple sine and cosine waves, and to write the [[superposition principle|solution as a superposition]] of the corresponding [[eigenfunction|eigensolutions]]. This superposition or linear combination is called the Fourier series.
From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of [[function (mathematics)|function]] and [[integral]] in the early nineteenth century. Later, [[Peter Gustav Lejeune Dirichlet]]<ref>Lejeune-Dirichlet, P. "[[List of important publications in mathematics#Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données|Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données]]". (In French), transl. "On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits". Journal f¨ur die reine und angewandte Mathematik, Vol. 4 (1829) p. 157–169.</ref> and [[Bernhard Riemann]]<ref>{{cite web|url = http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/| title=Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe |language=German|work=[[Habilitationschrift]], [[Göttingen]]; 1854. Abhandlungen der [[Göttingen Academy of Sciences|Königlichen Gesellschaft der Wissenschaften zu Göttingen]], vol. 13, 1867''. Published posthumously for Riemann by [[Richard Dedekind]]|trans_title=About the representability of a function by a trigonometric series|accessdate= 19 May 2008 <!--DASHBot-->| archiveurl= http://web.archive.org/web/20080520085248/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/ | archivedate= 20 May 2008| deadurl= no}}</ref><ref>D. Mascre, Bernhard Riemann: Posthumous Thesis on the Representation of Functions by Triginometric Series (1867). [http://books.google.co.uk/books?id=UdGBy8iLpocC&printsec=frontcover#v=onepage&q&f=false Landmark Writings in Western Mathematics 1640–1940], Ivor Grattan-Guinness (ed.); pg. 492. Elsevier, 20 May 2005.Accessed 7 Dec 2012.</</ref><ref>[http://books.google.co.uk/books?id=uP8SF4jf7GEC&printsec=frontcover#v=onepage&q&f=false Theory of Complex Functions: Readings in Mathematics], by Reinhold Remmert; pg 29. Springer, 1991. Accessed 7 Dec 2012.</ref> expressed Fourier's results with greater precision and formality.
Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are [[Sine wave|sinusoid]]s. The Fourier series has many such applications in [[electrical engineering]], [[oscillation|vibration]] analysis, [[acoustics]], [[optics]], [[signal processing]], [[image processing]], [[quantum mechanics]], [[econometrics]],<ref>{{cite book |first=Marc |last=Nerlove |first2=David M. |last2=Grether |first3=Jose L. |last3=Carvalho |year=1995 |title=Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics |location= |publisher=Elsevier |isbn=0-12-515751-7 }}</ref> [[Thin-shell structure|thin-walled shell]] theory,<ref>{{cite book |first=Wilhelm |last=Flugge |year=1957 |title=Statik und Dynamik der Schalen |publisher=Springer-Verlag |location=Berlin }}</ref> etc.
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