elliptic function

A meromorphic function is called an elliptic function, if there are two $\backslash mathbb\{R\}$-linear independent complex numbers $\backslash omega\_1,\backslash omega\_2\backslash in\backslash mathbb\{C\}$ such that

: $f(z\; +\; \backslash omega\_1)\; =\; f(z)$ and $f(z\; +\; \backslash omega\_2)\; =\; f(z),\; \backslash quad\; \backslash forall\; z\backslash in\backslash mathbb\{C\}$.

So elliptic functions have two periods and are therefore doubly periodic functions.

Image:Lattice torsion points.svg of its period lattice.]]

File:Torus from rectangle.gif

If $f$ is an elliptic function with periods $\backslash omega\_1,\backslash omega\_2$ it also holds that

: $f(z+\backslash gamma)=f(z)$

for every linear combination $\backslash gamma=m\backslash omega\_1+n\backslash omega\_2$ with $m,n\backslash in\backslash mathbb\{Z\}$.

The abelian group

: $\backslash Lambda:=\backslash langle\; \backslash omega\_1,\backslash omega\_2\backslash rangle\_\{\backslash mathbb\; Z\}:=\backslash mathbb\; Z\backslash omega\_1+\backslash mathbb\; Z\backslash omega\_2:=\backslash \{m\backslash omega\_1+n\backslash omega\_2\backslash mid\; m,n\backslash in\backslash mathbb\; Z\backslash \}$

is called the period lattice.

The parallelogram generated by $\backslash omega\_1$and $\backslash omega\_2$

: $\backslash \{\backslash mu\backslash omega\_1+\backslash nu\backslash omega\_2\backslash mid\; 0\backslash leq\backslash mu,\backslash nu\backslash leq\; 1\backslash \}$

is a fundamental domain of $\backslash Lambda$ acting on $\backslash C$.

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 $\backslash mathbb\{C\}/\backslash Lambda$ as their domain. This quotient group, called an elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus.{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 259|isbn=978-3-540-32058-6|date=2006|language=German

}}

The following three theorems are known as Liouville's theorems (1847).

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

}}

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

}} 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.

Every elliptic function has finitely many poles in $\backslash mathbb\{C\}/\backslash Lambda$ and the sum of its residues is zero.{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 260|isbn=978-3-540-32058-6|date=2006|language=German

}}

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 $\backslash mathbb\{C\}/\backslash Lambda$ counted with multiplicity.{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 262|isbn=978-3-540-32058-6|date=2006|language=German

}}

{{Main|Weierstrass elliptic function}}

One of the most important elliptic functions is the Weierstrass $\backslash wp$-function. For a given period lattice $\backslash Lambda$ it is defined by

: $\backslash wp(z)=\backslash frac1\{z^2\}+\backslash sum\_\{\backslash lambda\backslash in\backslash Lambda\backslash setminus\backslash \{0\backslash \}\}\backslash left(\backslash frac1\{(z-\backslash lambda)^2\}-\backslash frac1\{\backslash lambda^2\}\backslash right).$

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

$\backslash wp$ is an even elliptic function; that is, $\backslash wp(-z)=\backslash wp(z)$.{{citation|surname1=K. Chandrasekharan|title=Elliptic functions|publisher=Springer-Verlag|publication-place=Berlin|at=p. 28|isbn=0-387-15295-4|date=1985|language=German

}}

Its derivative

: $\backslash wp\text{'}(z)=-2\backslash sum\_\{\backslash lambda\backslash in\backslash Lambda\}\backslash frac1\{(z-\backslash lambda)^3\}$

is an odd function, i.e. $\backslash wp\text{'}(-z)=-\backslash wp\text{'}(z).$

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

}}

The $\backslash wp$-function satisfies the differential equation

: $\backslash wp\text{'}(z)^2=4\backslash wp(z)^3-g\_2\backslash wp(z)-g\_3,$

where $g\_2$ and $g\_3$ are constants that depend on $\backslash Lambda$. More precisely, $g\_2(\backslash omega\_1,\backslash omega\_2)=60G\_4(\backslash omega\_1,\backslash omega\_2)$ and $g\_3(\backslash omega\_1,\backslash omega\_2)=140G\_6(\backslash omega\_1,\backslash omega\_2)$, where $G\_4$ and $G\_6$ are so called Eisenstein series.{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 276|isbn=978-3-540-32058-6|date=2006|language=German

}}

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

: $\backslash mathbb\; C(X)[Y]/(Y^2-4X^3+g\_2X+g\_3)$,

where the isomorphism maps $\backslash wp$ to $X$ and $\backslash wp\text{'}$ to $Y$.

File:Weierstrass-p-1.jpg|Weierstrass $\backslash wp$-function with period lattice $\backslash Lambda=\backslash mathbb\{Z\}+e^\{2\backslash pi\; i/6\}\backslash mathbb\{Z\}$

File:Weierstrass-dp-1.jpg|Derivative of the $\backslash wp$-function

The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken on by Niels Henrik Abel and Carl Gustav Jacobi.

Abel discovered elliptic functions by taking the inverse function $\backslash varphi$ of the elliptic integral function

: $\backslash alpha(x)=\backslash int\_0^x\; \backslash frac\{dt\}\{\backslash sqrt\{(1-c^2t^2)(1+e^2t^2)\}\}$

with $x=\backslash varphi(\backslash alpha)$.{{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. 74|isbn=978-3-319-23715-2|language=German

}}

Additionally he defined the functions{{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. 75|isbn=978-3-319-23715-2|language=German

}}

: $f(\backslash alpha)=\backslash sqrt\{1-c^2\backslash varphi^2(\backslash alpha)\}$

and

: $F(\backslash alpha)=\backslash sqrt\{1+e^2\backslash varphi^2(\backslash alpha)\}$.

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

: $\backslash xi(x)=\backslash int\_0^x\; \backslash frac\{dt\}\{\backslash sqrt\{(1-t^2)(1-k^2t^2)\}\}$

and inverted it: $x=\backslash operatorname\{sn\}(\backslash xi)$. $\backslash operatorname\{sn\}$ stands for sinus amplitudinis and is the name of the new function.{{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. 82|isbn=978-3-319-23715-2|language=German

}} He then introduced the functions cosinus amplitudinis and delta amplitudinis, which are defined as follows:

: $\backslash operatorname\{cn\}(\backslash xi):=\backslash sqrt\{1-x^2\}$

: $\backslash operatorname\{dn\}(\backslash xi):=\backslash sqrt\{1-k^2x^2\}$

.

Only by taking this step, Jacobi could prove his general transformation formula of elliptic integrals in 1827.{{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. 81|isbn=978-3-319-23715-2|language=German

}}

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.{{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}} 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 LandenJohn 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. 283–289, {{JSTOR|106197}}. his ideas were not pursued until 1786, when Legendre published his paper Mémoires sur les intégrations par arcs d’ellipse.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. 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. 644–683. 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),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. 1–34. Exercices de calcul intégral (1811–1817),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. Traité des fonctions elliptiques (1825–1832).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. 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.Carl Gustav Jacob Jacobi: [https://books.google.com/books?id=wLKbL6-GwhUC Fundamenta nova theoriae functionum ellipticarum.] Königsberg 1829. 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 Briot and Bouquet in 1856.{{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}} Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.{{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}}

• Elliptic integral
• Elliptic curve
• Modular group
• Theta function

{{Reflist}}

{{Refbegin|30em}}

{{Refend}}

• {{Abramowitz_Stegun_ref2|16|567|18|627}} (only considers the case of real invariants).
• 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}}
• 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

{{Commons category|Elliptic functions}}

• {{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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