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{{Short description|Class of periodic mathematical functions}}
In the mathematical field of [[Complex analysis|complex analysis]], '''elliptic functions''' are special kinds of [[Meromorphic function|meromorphic]] functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from [[Elliptic integral|elliptic integral]]s. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an [[Ellipse|ellipse]].

Important elliptic functions are [[Jacobi elliptic functions]] and the [[Weierstrass's elliptic functions|Weierstrass <math>\wp</math>-function]].

Further development of this theory led to [[Hyperelliptic curve|hyperelliptic functions]] and [[Modular form|modular form]]s.

== Definition ==
A [[Meromorphic function|meromorphic function]] is called an elliptic function, if there are two  <math>\mathbb{R}</math>-[[Linear independence|linear independent]] [[Complex number|complex number]]s <math>\omega_1,\omega_2\in\mathbb{C}</math> such that

: <math>f(z + \omega_1) = f(z)</math> and <math>f(z + \omega_2) = f(z), \quad \forall z\in\mathbb{C}</math>.

So elliptic functions have two periods and are therefore [[Doubly periodic function|doubly periodic functions]].

== Period lattice and fundamental domain ==
[[File:Torus from rectangle.gif|thumb|Parallelogram where opposite sides are identified]]
If <math>f</math> is an elliptic function with periods <math>\omega_1,\omega_2</math> it also holds that

: <math>f(z+\gamma)=f(z)</math>

for every linear combination <math>\gamma=m\omega_1+n\omega_2</math> with <math>m,n\in\mathbb{Z} </math>.

The [[Abelian group|abelian group]]

: <math>\Lambda:=\langle \omega_1,\omega_2\rangle_{\mathbb Z}:=\mathbb Z\omega_1+\mathbb Z\omega_2:=\{m\omega_1+n\omega_2\mid m,n\in\mathbb Z\}</math>

is called the ''period lattice''.

The [[Parallelogram|parallelogram]] generated by <math>\omega_1</math>and <math>\omega_2</math>

: <math>\{\mu\omega_1+\nu\omega_2\mid 0\leq\mu,\nu\leq 1\}</math>

is a ''[[Fundamental domain|fundamental domain]]'' of <math>\Lambda</math> [[Group action|acting]] on <math>\C</math>.

Geometrically the complex plane is tiled with parallelograms. Everything that happens in one fundamental domain repeats in all the others. For that reason we can view elliptic function as functions with the [[Quotient group|quotient group]] <math>\mathbb{C}/\Lambda</math> as their domain. This quotient group, called an [[Elliptic curve|elliptic curve]], can be visualised as a parallelogram where opposite sides are identified, which [[Topology|topologically]] is a [[Torus|torus]].<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;259|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

== Liouville's theorems ==
The following three theorems are known as ''[[Biography:Joseph Liouville|Liouville]]'s theorems (1847).''

=== 1st theorem ===
A holomorphic elliptic function is constant.<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;258|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

This is the original form of [[Liouville's theorem (complex analysis)|Liouville's theorem]] and can be derived from it.<ref>{{citation|surname1=Jeremy Gray|title=Real and the complex : a history of analysis in the 19th century|publication-place=Cham|at=pp.&nbsp;118f|isbn=978-3-319-23715-2|date=2015|language=German
}}</ref>  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.

=== 2nd theorem ===
Every elliptic function has finitely many poles in <math>\mathbb{C}/\Lambda</math> and the sum of its [[Residue (complex analysis)|residues]] is zero.<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;260|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

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.

=== 3rd theorem ===
A non-constant elliptic function takes on every value the same number of times in <math>\mathbb{C}/\Lambda</math> counted with multiplicity.<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;262|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

==Weierstrass ℘-function==
{{Main|Weierstrass elliptic function}}

One of the most important elliptic functions is the Weierstrass <math>\wp</math>-function. For a given period lattice <math>\Lambda</math> it is defined by

: <math>\wp(z)=\frac1{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(z-\lambda)^2}-\frac1{\lambda^2}\right).</math>

It is constructed in such a way that it has a pole of order two at every lattice point. The term <math>-\frac1{\lambda^2}</math> is there to make the series convergent.

<math>\wp</math> is an even elliptic function; that is, <math>\wp(-z)=\wp(z)</math>.<ref name=":0">{{citation|surname1=K. Chandrasekharan|title=Elliptic functions|publisher=Springer-Verlag|publication-place=Berlin|at=p.&nbsp;28|isbn=0-387-15295-4|date=1985|language=German
}}</ref>

Its derivative

: <math>\wp'(z)=-2\sum_{\lambda\in\Lambda}\frac1{(z-\lambda)^3}</math>

is an odd function, i.e. <math>\wp'(-z)=-\wp'(z).</math><ref name=":0" />

One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice <math>\Lambda</math> can be expressed as a rational function in terms of <math>\wp</math> and <math>\wp'</math>.<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;275|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

The <math>\wp</math>-function satisfies the [[Differential equation|differential equation]]

: <math>\wp'^2(z)=4\wp(z)^3-g_2\wp(z)-g_3,</math>

where <math>g_2</math> and <math>g_3</math> are constants that depend on <math>\Lambda</math>. More precisely, <math>g_2(\omega_1,\omega_2)=60G_4(\omega_1,\omega_2)</math> and <math>g_3(\omega_1,\omega_2)=140G_6(\omega_1,\omega_2)</math>, where <math>G_4</math> and <math>G_6</math> are so called [[Eisenstein series]].<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;276|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

In algebraic language, the field of elliptic functions is isomorphic to the field

: <math>\mathbb C(X)[Y]/(Y^2-4X^3+g_2X+g_3)</math>,

where the isomorphism maps <math>\wp</math> to <math>X</math> and <math>\wp'</math> to <math>Y</math>.<gallery>
File:Weierstrass-p-1.jpg|Weierstrass <math>\wp</math>-function with period lattice <math>\Lambda=\mathbb{Z}+e^{2\pi i/6}\mathbb{Z}  </math>
File:Weierstrass-dp-1.jpg|Derivative of the <math>\wp</math>-function
</gallery>

== Relation to elliptic integrals ==
The relation to [[Elliptic integral|elliptic integrals]] has mainly a historical background. Elliptic integrals had been studied by [[Biography:Adrien-Marie Legendre|Legendre]], whose work was taken on by [[Biography:Niels Henrik Abel|Niels Henrik Abel]] and [[Biography:Carl Gustav Jacob Jacobi|Carl Gustav Jacobi]].

Abel discovered elliptic functions by taking the inverse function <math>\varphi</math> of the elliptic integral function

: <math>\alpha(x)=\int_0^x \frac{dt}{\sqrt{(1-c^2t^2)(1+e^2t^2)}}</math>

with <math>x=\varphi(\alpha)</math>.<ref>{{citation|surname1=Gray, Jeremy|title=Real and the complex : a history of analysis in the 19th century|date=14 October 2015|publication-place=Cham|at=p.&nbsp;74|isbn=978-3-319-23715-2|language=German
}}</ref>

Additionally he defined the functions<ref>{{citation|surname1=Gray, Jeremy|title=Real and the complex : a history of analysis in the 19th century|date=14 October 2015|publication-place=Cham|at=p.&nbsp;75|isbn=978-3-319-23715-2|language=German
}}</ref>

: <math>f(\alpha)=\sqrt{1-c^2\varphi^2(\alpha)}</math>

and

: <math>F(\alpha)=\sqrt{1+e^2\varphi^2(\alpha)}</math>.

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

: <math>\xi(x)=\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}</math>

and inverted it: <math>x=\operatorname{sn}(\xi)</math>. <math>\operatorname{sn}</math> stands for ''sinus amplitudinis'' and is the name of the new function.<ref>{{citation|surname1=Gray, Jeremy|title=Real and the complex : a history of analysis in the 19th century|date=14 October 2015|publication-place=Cham|at=p.&nbsp;82|isbn=978-3-319-23715-2|language=German
}}</ref> He then introduced the functions ''cosinus amplitudinis''  and ''delta amplitudinis'', which are defined as follows:

: <math>\operatorname{cn}(\xi):=\sqrt{1-x^2} 
  </math>

: <math>\operatorname{dn}(\xi):=\sqrt{1-k^2x^2}
 
  </math>.

Only by taking this step, Jacobi could prove his general transformation formula of elliptic integrals in 1827.<ref>{{citation|surname1=Gray, Jeremy|title=Real and the complex : a history of analysis in the 19th century|date=14 October 2015|publication-place=Cham|at=p.&nbsp;81|isbn=978-3-319-23715-2|language=German
}}</ref>

== History ==
Shortly after the development of [[Calculus|infinitesimal calculus]] the theory of elliptic functions was started by the Italian mathematician Giulio di Fagnano and the Swiss mathematician [[Biography:Leonhard Euler|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 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 [[Biography: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, [[Biography:Niels Henrik Abel|Niels Henrik Abel]] and [[Biography: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. 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 Briot and 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> [[Biography: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>

==See also==
* [[Elliptic integral]]
* [[Elliptic curve]]
* [[Modular group]]
* [[Theta function]]

==References==

{{Reflist}}
{{Refbegin|30em}}
{{Refend}}

==Literature==
* {{Abramowitz_Stegun_ref2|16|567|18|627}} (only considers the case of real invariants).
* [[Biography:Naum Akhiezer|N. I. Akhiezer]], ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island {{isbn|0-8218-4532-2}}
* [[Biography:Tom M. Apostol|Tom M. Apostol]], ''Modular Functions and Dirichlet Series in Number Theory'', Springer-Verlag, New York, 1976. {{isbn|0-387-97127-0}} ''(See Chapter 1.)''
* [[Biography:E. T. Whittaker|E. T. Whittaker]] and [[Biography:G. N. Watson|G. N. Watson]]. ''[[Whittaker and Watson|A course of modern analysis]]'', Cambridge University Press, 1952

==External links==

* {{springer|title=Elliptic function|id=p/e035470}}
* MAA, [http://www.maa.org/press/periodicals/convergence/abel-on-elliptic-integrals-a-translation  Translation of Abel's paper on elliptic functions.] 
* {{YouTube|DCXItCajCyo|title=Elliptic Functions and Elliptic Integrals}}, lecture by William A. Schwalm (4 hours)
* {{cite arXiv |eprint=1806.06725|last1=Johansson|first1=Fredrik|title=Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms|class=cs.NA|year=2018}}

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