In the mathematical field of complex analysis elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse.
- and .
So elliptic functions have two periods and are therefore also called doubly periodic.
Period lattice and fundamental domain
If is an elliptic function with periods it also holds that
for every linear combination with .
The abelian group
is called the period lattice.
The parallelogram generated by and
is called fundamental domain.
Geometrically the complex plane is tiled with parallelograms. Everything that happens in the fundamental domain repeats in all the others. For that reason we can view elliptic function as functions with the quotient group as their domain. This quotient group can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus.
The following three theorems are known as Liouville's theorems (1847).
A holomorphic elliptic function is constant.
This is the original form of Liouville's theorem and can be derived from it. A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's theorem.
This theorem implies that there is no elliptic function not equal to zero with exactly one pole of order one or exactly one zero of order one in the fundamental domain.
A non-constant elliptic function takes on every value the same number of times in counted with multiplicity.
One of the most important elliptic functions is the Weierstrass -function. For a given period lattice it is defined by
It is constructed in such a way that it has a pole of order two at every lattice point. The term is there to make the series convergent.
is an even elliptic function, that means .
is an odd function, i.e. 
One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice can be expressed as a rational function in terms of and .
The -function satisfies the differential equation
In algebraic language: The field of elliptic functions is isomorphic to the field
where the isomorphism maps to and to .
Relation to elliptic Integrals
Abel discovered elliptic functions by taking the inverse function of the elliptic integral function
Additionally he defined the functions
After continuation to the complex plane they turned out to be doubly periodic and are known as Abel elliptic functions.
Jacobi elliptic functions are similarly obtained as inverse functions of elliptic integrals.
Jacobi considered the integral function
and inverted it: . stands for sinus amplitudinis and is the name of the new function. He then introduced the functions cosinus amplitudinis and delta amplitudinis, which are defined as follows:
Only by taking this step, Jacobi could prove his general transformation formula of elliptic integrals in 1827.
Shortly after the development of infinitesimal calculus the theory of elliptic functions was started by the Italian mathematician 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. 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. Euler immediately generalized Fagnano's results and posed his algebraic addition theorem for elliptic integrals.
Except for a comment by Landen his ideas were not pursued until 1786, when Legendre published his paper Mémoires sur les intégrations par arcs d’ellipse. 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), Exercices de calcul intégral (1811–1817), Traité des fonctions elliptiques (1825–1832). Legendre's work was mostly left untouched by mathematicians until 1826.
Subsequently, Niels Henrik Abel and 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. The addition theorem Euler found was posed and proved in its general form by Abel in 1829. Note that 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 Briout and Bouquet in 1856. Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.
- Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN 978-3-540-32058-6
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- Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 276, ISBN 978-3-540-32058-6
- Gray, Jeremy (14 October 2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 74, ISBN 978-3-319-23715-2
- Gray, Jeremy (14 October 2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 75, ISBN 978-3-319-23715-2
- Gray, Jeremy (14 October 2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 82, ISBN 978-3-319-23715-2
- Gray, Jeremy (14 October 2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 81, ISBN 978-3-319-23715-2
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- Adrien-Marie Legendre: 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 A Memoire on Elliptic Transcendentals. In: Thomas Leybourn: New Series of the Mathematical Repository. Band 2. Glendinning, London 1809, Teil 3, S. 1–34.
- Adrien-Marie Legendre: Exercices de calcul integral sur divers ordres de transcendantes et sur les quadratures. 3 Bände. (Band 1, Band 2, Band 3). Paris 1811–1817.
- Adrien-Marie Legendre: Traité des fonctions elliptiques et des intégrales eulériennes, avec des tables pour en faciliter le calcul numérique. 3 Bde. (Band 1, Band 2, Band 3/1, Band 3/2, Band 3/3). Huzard-Courcier, Paris 1825–1832.
- Carl Gustav Jacob Jacobi: Fundamenta nova theoriae functionum ellipticarum. Königsberg 1829.
- Gray, Jeremy (2015). Real and the complex : a history of analysis in the 19th century. Cham. p. 122. ISBN 978-3-319-23715-2. OCLC 932002663.
- Gray, Jeremy (2015). Real and the complex : a history of analysis in the 19th century. Cham. p. 96. ISBN 978-3-319-23715-2. OCLC 932002663.
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- Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976. ISBN 0-387-97127-0 (See Chapter 1.)
- E. T. Whittaker and G. N. Watson. A course of modern analysis, Cambridge University Press, 1952
|Wikimedia Commons has media related to Elliptic functions.|
- "Elliptic function", Encyclopedia of Mathematics, EMS Press, 2001 
- MAA, Translation of Abel's paper on elliptic functions.
- on YouTube, lecture by William A. Schwalm (4 hours)
- Johansson, Fredrik (2018). "Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms". arXiv:1806.06725 [cs.NA].