Editing Fourier series (section)

==History{{anchor|Historical development}}==
{{See also|Fourier analysis#History}}

The Fourier series is named in honor 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]].{{efn-ua|
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 theory|comtemporaneous/ in collaboration with Bernoulli]], although the latter made some independent contributions to the theory of waves and vibrations. (See {{harvnb|Fetter|Walecka|2003|pp=209–210}}).
}} 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'' (''Analytical theory of heat'') in 1822. The ''Mémoire'' introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous<ref name="Stillwell2013">{{cite book|last=Stillwell|first=John|title=Routledge History of Philosophy|publisher=Routledge|year=2013|isbn=978-1-134-92880-4|editor-last=Ten|editor-first=C. L.|volume=VII: The Nineteenth Century|page=204|chapter=Logic and the philosophy of mathematics in the nineteenth century|author-link=John Stillwell|chapter-url=https://books.google.com/books?id=91AqBgAAQBAJ&pg=PA204}}</ref> and later generalized to any [[piecewise]]-smooth<ref name="iit.edu" />) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the [[Académie française|French Academy]].<ref name="Cajori1893">{{cite book |last=Cajori |first=Florian |author-link=Florian Cajori |url=https://archive.org/details/ahistorymathema00cajogoog |title=A History of Mathematics |publisher=Macmillan |year=1893 |page=[https://archive.org/details/ahistorymathema00cajogoog/page/n303 283]}}</ref> 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]].

Independently of Fourier, astronomer [[Friedrich Wilhelm Bessel]] introduced Fourier series to solve [[Kepler's equation]]. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.<ref>{{cite journal |last1=Dutka |first1=Jacques |date=1995 |title=On the early history of Bessel functions |journal=Archive for History of Exact Sciences |volume=49 |issue=2 |pages=105–134 |doi=10.1007/BF00376544}}</ref>

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>{{cite journal|last=Lejeune-Dirichlet|first=Peter Gustav|author-link=Peter Gustav Lejeune Dirichlet|year=1829|title=Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données|trans-title=On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits|url=https://archive.org/details/arxiv-0806.1294|journal=[[Journal für die reine und angewandte Mathematik]]|language=fr|volume=4|pages=157–169|arxiv=0806.1294}}</ref> and [[Bernhard Riemann]]<ref>{{cite web|title=Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe|trans-title=About the representability of a function by a trigonometric series|url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/|url-status=live|archive-url=https://web.archive.org/web/20080520085248/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/|archive-date=20 May 2008|access-date=19 May 2008|work=[[Habilitationsschrift]], [[Göttingen]]; 1854. Abhandlungen der [[Göttingen Academy of Sciences|Königlichen Gesellschaft der Wissenschaften zu Göttingen]], vol. 13, 1867. Published posthumously for Riemann by [[Richard Dedekind]]|language=de}}</ref><ref>{{citation|last1=Mascre|first1=D.|title=Landmark Writings in Western Mathematics 1640–1940|date=1867|url=https://books.google.com/books?id=UdGBy8iLpocC|page=49|publication-date=2005|editor-last=Grattan-Guinness|editor-first=Ivor|chapter=Posthumous Thesis on the Representation of Functions by Trigonometric Series|publisher=Elsevier|isbn=9780080457444|last2=Riemann|first2=Bernhard}}</ref><ref>{{cite book|last=Remmert|first=Reinhold|url=https://books.google.com/books?id=uP8SF4jf7GEC|title=Theory of Complex Functions: Readings in Mathematics|date=1991|publisher=Springer|isbn=9780387971957|page=29}}</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|last1=Nerlove|first1=Marc|url=https://archive.org/details/analysisofeconom0000nerl|title=Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics|last2=Grether|first2=David M.|last3=Carvalho|first3=Jose L.|publisher=Elsevier|year=1995|isbn=0-12-515751-7|url-access=registration}}</ref> [[Membrane theory of shells|shell theory]],<ref>[[Wilhelm Flügge]], ''Stresses in Shells'' (1973) 2nd edition. {{isbn|978-3-642-88291-3}}. Originally published in German as ''Statik und Dynamik der Schalen'' (1937).</ref> etc.

=== Beginnings ===
Joseph Fourier wrote<ref>{{cite book |title=Oeuvres de Fourier|date= 1890| pages= 218–219|last= Fourier|first= Jean-Baptiste-Joseph|authorlink= Jean-Baptiste-Joseph Fourier |editor=Gaston Darboux|chapter=Mémoire sur la propagation de la chaleur dans les corps solides, présenté le 21 Décembre 1807 à l'Institut national| publisher=Gauthier-Villars et Fils|location=Paris |language=fr |isbn=9781139568159|volume=2|trans-title=The Works of Fourier |doi=10.1017/CBO9781139568159.009}} {{pb}} Whilst the cited article does list the author as Fourier, a footnote on page 215 indicates that the article was actually written by [[Siméon_Denis_Poisson|Poisson]] and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.</ref>

{{blockquote|<math>\varphi(y)=a_0\cos\frac{\pi y}{2}+a_1\cos 3\frac{\pi y}{2}+a_2\cos5\frac{\pi y}{2}+\cdots.</math>

Multiplying both sides by <math>\cos(2k+1)\frac{\pi y}{2}</math>, and then integrating from <math>y=-1</math> to <math>y=+1</math> yields:

<math>a_k=\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy.</math>
|Joseph Fourier|[[Mémoire sur la propagation de la chaleur dans les corps solides]] (1807).}}

This immediately gives any coefficient ''a<sub>k</sub>'' of the [[trigonometric series]] for φ(''y'') for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral
<math display="block">\begin{align}
&\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy \\
&= \int_{-1}^1\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy
\end{align}</math>
can be carried out term-by-term. But all terms involving <math>\cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}</math> for {{nowrap|''j'' &ne; ''k''}} vanish when integrated from −1 to 1, leaving only the <math>k^{\text{th}}</math> term, which is ''1''.

In these few lines, which are close to the modern [[Formalism (mathematics)|formalism]] used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by [[Euler]], [[Jean le Rond d'Alembert|d'Alembert]], [[Daniel Bernoulli]] and [[Carl Friedrich Gauss|Gauss]], Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of [[Convergent series|convergence]], [[function space]]s, and [[harmonic analysis]].

When Fourier submitted a later competition essay in 1811, the committee (which included [[Joseph Louis Lagrange|Lagrange]], [[Laplace]], [[Étienne-Louis Malus|Malus]] and [[Adrien-Marie Legendre|Legendre]], among others) concluded: ''...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even [[Mathematical rigour|rigour]]''.<ref>{{cite book |title=Oeuvres de Fourier|date= 1888|last= Fourier|first= Jean-Baptiste-Joseph|authorlink= Jean-Baptiste-Joseph Fourier |editor=Gaston Darboux|chapter=Avant-propos des oevres de Fourier| publisher=Gauthier-Villars et Fils|location=Paris |language=fr |volume=1|trans-title=The Works of Fourier  | isbn=978-1-108-05938-1 | doi=10.1017/cbo9781139568081.001 |pages=VII-VIII}}</ref>

===Fourier's motivation===
[[File:Fourier heat in a plate.png|thumb|right|This resulting heat distribution in a metal plate is easily solved using Fourier's method]]
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula <math>s(x)=\tfrac{x}{\pi}</math>, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the [[heat equation]]. For example, consider a metal plate in the shape of a square whose sides measure <math>\pi</math> meters, with coordinates <math>(x,y) \in [0,\pi] \times [0,\pi]</math>. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by <math>y=\pi</math>, is maintained at the temperature gradient <math>T(x,\pi)=x</math> degrees Celsius, for <math>x</math> in <math>(0,\pi)</math>, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by
: <math>T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.</math>
Here, sinh is the [[hyperbolic sine]] function. This solution of the heat equation is obtained by multiplying each term of the equation from [[#Example|Analysis § Example]] by <math>\sinh(ny)/\sinh(n\pi)</math>. While our example function <math>s(x)</math> seems to have a needlessly complicated Fourier series, the heat distribution <math>T(x,y)</math> is nontrivial. The function <math>T</math> cannot be written as a [[closed-form expression]]. This method of solving the heat problem was made possible by Fourier's work.

===Other applications===
Another application is to solve the [[Basel problem]] by using [[Parseval's theorem]]. The example generalizes and one may compute [[Riemann zeta function|ζ]](2''n''), for any positive integer ''n''.