Editing Elliptic function (section)

== History ==
Shortly after the development of [[Calculus|infinitesimal calculus]] the theory of elliptic functions was started by the Italian mathematician [[Giulio Carlo de' Toschi di Fagnano|Giulio di Fagnano]] and the Swiss mathematician [[Leonhard Euler]]. When they tried to calculate the arc length of a [[lemniscate]] they encountered problems involving integrals that contained the square root of polynomials of degree 3 and 4.<ref name=":1">{{Cite book|last=Gray|first=Jeremy|url=https://www.worldcat.org/oclc/932002663|title=Real and the complex : a history of analysis in the 19th century|date=2015|isbn=978-3-319-23715-2|location=Cham|pages=23f|oclc=932002663}}</ref> It was clear that those so called elliptic integrals could not be solved using elementary functions. Fagnano observed an algebraic relation between elliptic integrals, what he published in 1750.<ref name=":1" /> Euler immediately generalized Fagnano's results and posed his algebraic addition theorem for elliptic integrals.<ref name=":1" />

Except for a comment by [[John Landen|Landen]]<ref>John Landen: ''An Investigation of a general Theorem for finding the Length of any Arc of any Conic Hyperbola, by Means of Two Elliptic Arcs, with some other new and useful Theorems deduced therefrom.'' In: ''The Philosophical Transactions of the Royal Society of London'' 65 (1775), Nr. XXVI, S.&nbsp;283–289, {{JSTOR|106197}}.</ref> his ideas were not pursued until 1786, when [[Adrien-Marie Legendre|Legendre]] published his paper ''Mémoires sur les intégrations par arcs d’ellipse''.<ref>Adrien-Marie Legendre: [https://books.google.com/books?id=rIYlBNp4oiIC&pg=616 ''Mémoire sur les intégrations par arcs d’ellipse.''] In: ''Histoire de l’Académie royale des sciences Paris'' (1788), S.&nbsp;616–643. – Ders.: [https://books.google.com/books?id=rIYlBNp4oiIC&pg=644 ''Second mémoire sur les intégrations par arcs d’ellipse, et sur la comparaison de ces arcs.''] In: ''Histoire de l’Académie royale des sciences Paris'' (1788), S.&nbsp;644–683.</ref> Legendre subsequently studied elliptic integrals and called them ''elliptic functions''. Legendre introduced a three-fold classification – three kinds – which was a crucial simplification of the rather complicated theory at that time. Other important works of Legendre are: ''Mémoire sur les transcendantes elliptiques'' (1792),<ref>Adrien-Marie Legendre: [https://books.google.com/books?id=tR3pvoE3HcMC ''Mémoire sur les transcendantes elliptiques''], ''où l’on donne des méthodes faciles pour comparer et évaluer ces trancendantes, qui comprennent les arcs d’ellipse, et qui se rencontrent frèquemment dans les applications du calcul intégral.'' Du Pont & Firmin-Didot, Paris 1792. Englische Übersetzung [https://books.google.com/books?id=vNULAAAAYAAJ&pg=347 ''A Memoire on Elliptic Transcendentals.''] In: Thomas Leybourn: ''New Series of the Mathematical Repository''. Band 2. Glendinning, London 1809, Teil 3, S.&nbsp;1–34.</ref> ''Exercices de calcul intégral'' (1811–1817),<ref>Adrien-Marie Legendre: ''Exercices de calcul integral sur divers ordres de transcendantes et sur les quadratures.'' 3 Bände. ([https://books.google.com/books?id=riIOAAAAQAAJ Band 1], [https://books.google.com/books?id=6yIOAAAAQAAJ Band 2], Band 3). Paris 1811–1817.</ref> ''Traité des fonctions elliptiques'' (1825–1832).<ref>Adrien-Marie Legendre: ''Traité des fonctions elliptiques et des intégrales eulériennes, avec des tables pour en faciliter le calcul numérique.'' 3 Bde. ([https://books.google.com/books?id=0iAOAAAAQAAJ&pg=PR3 Band 1], [https://books.google.com/books?id=UZIKAAAAYAAJ Band 2], [https://books.google.com/books?id=2ZIKAAAAYAAJ&pg=PR3 Band 3/1], Band 3/2, Band 3/3). Huzard-Courcier, Paris 1825–1832.</ref> Legendre's work was mostly left untouched by mathematicians until 1826.

Subsequently, [[Niels Henrik Abel]] and [[Carl Gustav Jacob Jacobi|Carl Gustav Jacobi]] resumed the investigations and quickly discovered new results. At first they inverted the elliptic integral function. Following a suggestion of Jacobi in 1829 these inverse functions are now called ''elliptic functions''. One of Jacobi's most important works is ''Fundamenta nova theoriae functionum ellipticarum'' which was published 1829.<ref>Carl Gustav Jacob Jacobi: [https://books.google.com/books?id=wLKbL6-GwhUC ''Fundamenta nova theoriae functionum ellipticarum.''] Königsberg 1829.</ref> The addition theorem Euler found was posed and proved in its general form by Abel in 1829. In those days the theory of elliptic functions and the theory of doubly periodic functions were considered to be different theories. They were brought together by [[Charles Auguste Briot|Briot]] and [[Jean Claude Bouquet|Bouquet]] in 1856.<ref>{{Cite book|last=Gray|first=Jeremy|url=https://www.worldcat.org/oclc/932002663|title=Real and the complex : a history of analysis in the 19th century|date=2015|isbn=978-3-319-23715-2|location=Cham|pages=122|oclc=932002663}}</ref> [[Carl Friedrich Gauss|Gauss]] discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.<ref>{{Cite book|last=Gray|first=Jeremy|url=https://www.worldcat.org/oclc/932002663|title=Real and the complex : a history of analysis in the 19th century|date=2015|isbn=978-3-319-23715-2|location=Cham|pages=96|oclc=932002663}}</ref>