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Elliptic function

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Class of periodic mathematical functions

In the mathematical field ofcomplex analysis,elliptic functions are special kinds ofmeromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come fromelliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of anellipse.

Important elliptic functions areJacobi elliptic functions and theWeierstrass $\wp$ -function.

Further development of this theory led tohyperelliptic functions andmodular forms.

Definition

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Ameromorphic function is called an elliptic function, if there are two $\mathbb {R}$ -linear independent complex numbers $\omega _{1},\omega _{2}\in \mathbb {C}$ such that

f(z+\omega _{1})=f(z)

and

f(z+\omega _{2})=f(z),\quad \forall z\in \mathbb {C}

So elliptic functions have two periods and are thereforedoubly periodic functions.

Period lattice and fundamental domain

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The fundamental domain of an elliptic function as theunit cell of its period lattice.

Parallelogram where opposite sides are identified

If $f {\displaystyle f}$ is an elliptic function with periods $\omega _{1},\omega _{2}$ it also holds that

f(z+\gamma )=f(z)

for every linear combination $\gamma =m\omega _{1}+n\omega _{2}$ with $m,n\in \mathbb {Z}$ .

Theabelian group

\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} \}

is called theperiod lattice.

Theparallelogram generated by $\omega _{1}$ and $\omega _{2}$

\{\mu \omega _{1}+\nu \omega _{2}\mid 0\leq \mu ,\nu \leq 1\}

is afundamental domain of $\Lambda$ acting on $\mathbb {C}$ .

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 thequotient group $\mathbb {C} /\Lambda$ as their domain. This quotient group, called anelliptic curve, can be visualised as a parallelogram where opposite sides are identified, whichtopologically is atorus.^[1]

Liouville's theorems

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

1st theorem

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A holomorphic elliptic function is constant.^[2]

This is the original form ofLiouville's theorem and can be derived from it.^[3] 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

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Every elliptic function has finitely many poles in $\mathbb {C} /\Lambda$ and the sum of itsresidues is zero.^[4]

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

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A non-constant elliptic function takes on every value the same number of times in $\mathbb {C} /\Lambda$ counted with multiplicity.^[5]

Weierstrass ℘-function

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Main article:Weierstrass elliptic function

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

\wp (z)={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right).

It is constructed in such a way that it has a pole of order two at every lattice point. The term $-{\frac {1}{\lambda ^{2}}}$ is there to make the series convergent.

$\wp$ is an even elliptic function; that is, $\wp (-z)=\wp (z)$ .^[6]

Its derivative

\wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}

is an odd function, i.e. $\wp '(-z)=-\wp '(z).$ ^[6]

One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice $\Lambda$ can be expressed as a rational function in terms of $\wp$ and $\wp '$ .^[7]

The $\wp$ -function satisfies thedifferential equation

\wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3},

where $g_{2}$ and $g_{3}$ are constants that depend on $\Lambda$ . More precisely, $g_{2}(\omega _{1},\omega _{2})=60G_{4}(\omega _{1},\omega _{2})$ and $g_{3}(\omega _{1},\omega _{2})=140G_{6}(\omega _{1},\omega _{2})$ , where $G_{4}$ and $G_{6}$ are so calledEisenstein series.^[8]

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

\mathbb {C} (X)[Y]/(Y^{2}-4X^{3}+g_{2}X+g_{3})

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

Weierstrass $\wp$ -function with period lattice $\Lambda =\mathbb {Z} +e^{2\pi i/6}\mathbb {Z}$
Derivative of the $\wp$ -function

Relation to elliptic integrals

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The relation toelliptic integrals has mainly a historical background. Elliptic integrals had been studied byLegendre, whose work was taken on byNiels Henrik Abel andCarl Gustav Jacobi.

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

\alpha (x)=\int _{0}^{x}{\frac {dt}{\sqrt {(1-c^{2}t^{2})(1+e^{2}t^{2})}}}

with $x=\varphi (\alpha )$ .^[9]

Additionally he defined the functions^[10]

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

and

F(\alpha )={\sqrt {1+e^{2}\varphi ^{2}(\alpha )}}

After continuation to the complex plane they turned out to be doubly periodic and are known asAbel elliptic functions.

Jacobi elliptic functions are similarly obtained as inverse functions of elliptic integrals.

Jacobi considered the integral function

\xi (x)=\int _{0}^{x}{\frac {dt}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}

and inverted it: $x=\operatorname {sn} (\xi )$ . $\operatorname {sn}$ stands forsinus amplitudinis and is the name of the new function.^[11] He then introduced the functionscosinus amplitudinis anddelta amplitudinis, which are defined as follows:

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

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

Only by taking this step, Jacobi could prove his general transformation formula of elliptic integrals in 1827.^[12]

History

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Shortly after the development ofinfinitesimal calculus the theory of elliptic functions was started by the Italian mathematicianGiulio di Fagnano and the Swiss mathematicianLeonhard Euler. When they tried to calculate the arc length of alemniscate they encountered problems involving integrals that contained the square root of polynomials of degree 3 and 4.^[13] 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.^[13] Euler immediately generalized Fagnano's results and posed his algebraic addition theorem for elliptic integrals.^[13]

Except for a comment byLanden^[14] his ideas were not pursued until 1786, whenLegendre published his paperMémoires sur les intégrations par arcs d’ellipse.^[15] Legendre subsequently studied elliptic integrals and called themelliptic 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),^[16]Exercices de calcul intégral (1811–1817),^[17]Traité des fonctions elliptiques (1825–1832).^[18] Legendre's work was mostly left untouched by mathematicians until 1826.

Subsequently,Niels Henrik Abel andCarl 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 calledelliptic functions. One of Jacobi's most important works isFundamenta nova theoriae functionum ellipticarum which was published 1829.^[19] 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 byBriot andBouquet in 1856.^[20]Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.^[21]

References

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^Rolf Busam (2006),Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259,ISBN 978-3-540-32058-6
^Rolf Busam (2006),Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 258,ISBN 978-3-540-32058-6
^Jeremy Gray (2015),Real and the complex : a history of analysis in the 19th century (in German), Cham, pp. 118f,ISBN 978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
^Rolf Busam (2006),Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 260,ISBN 978-3-540-32058-6
^Rolf Busam (2006),Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 262,ISBN 978-3-540-32058-6
^^a ^bK. Chandrasekharan (1985),Elliptic functions (in German), Berlin: Springer-Verlag, p. 28,ISBN 0-387-15295-4
^Rolf Busam (2006),Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 275,ISBN 978-3-540-32058-6
^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{{citation}}: CS1 maint: location missing publisher (link)
^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{{citation}}: CS1 maint: location missing publisher (link)
^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{{citation}}: CS1 maint: location missing publisher (link)
^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{{citation}}: CS1 maint: location missing publisher (link)
^^a ^b ^cGray, Jeremy (2015).Real and the complex : a history of analysis in the 19th century. Cham. pp. 23f.ISBN 978-3-319-23715-2.OCLC 932002663.{{cite book}}: CS1 maint: location missing publisher (link)
^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. 283–289,JSTOR 106197.
^Adrien-Marie Legendre:Mémoire sur les intégrations par arcs d’ellipse. In:Histoire de l’Académie royale des sciences Paris (1788), S. 616–643. – Ders.: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.
^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 ÜbersetzungA 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.{{cite book}}: CS1 maint: location missing publisher (link)
^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.{{cite book}}: CS1 maint: location missing publisher (link)

Literature

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Abramowitz, Milton;Stegun, Irene Ann, eds. (1983) [June 1964]."Chapter 16".Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 567, 627.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253. See alsochapter 18. (only considers the case of real invariants).
N. I. Akhiezer,Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English asAMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode IslandISBN 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 andG. N. Watson.A course of modern analysis, Cambridge University Press, 1952

External links

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Wikimedia Commons has media related toElliptic functions.

"Elliptic function",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
MAA,Translation of Abel's paper on elliptic functions.
Elliptic Functions and Elliptic Integrals onYouTube, 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].

Topics inalgebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	A_k singularity Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves