In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse.
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 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.[1]
Liouville's theorems
The following three theorems are known as Liouville's theorems (1847).
This is the original form of Liouville'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
Every elliptic function has finitely many poles in and the sum of its residues 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
A non-constant elliptic function takes on every value the same number of times in counted with multiplicity.[5]
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 .[7]
and inverted it: . stands for sinus amplitudinis and is the name of the new function.[11] 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.[12]
History
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.[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 by Landen[14] his ideas were not pursued until 1786, when Legendre published his paper Mémoires sur les intégrations par arcs d’ellipse.[15] 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),[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 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.[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 by Briot and Bouquet in 1856.[20]Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.[21]
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259, ISBN978-3-540-32058-6
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 258, ISBN978-3-540-32058-6
^Jeremy Gray (2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, pp. 118f, ISBN978-3-319-23715-2{{citation}}: CS1 maint: location missing publisher (link)
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 260, ISBN978-3-540-32058-6
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 262, ISBN978-3-540-32058-6
^ abK. Chandrasekharan (1985), Elliptic functions (in German), Berlin: Springer-Verlag, p. 28, ISBN0-387-15295-4
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 275, ISBN978-3-540-32058-6
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 276, ISBN978-3-540-32058-6
^Gray, Jeremy (14 October 2015), Real and the complex : a history of analysis in the 19th century (in German), Cham, p. 74, ISBN978-3-319-23715-2{{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, ISBN978-3-319-23715-2{{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, ISBN978-3-319-23715-2{{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, ISBN978-3-319-23715-2{{citation}}: 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, JSTOR106197.
^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.
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 ISBN0-8218-4532-2
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976. ISBN0-387-97127-0(See Chapter 1.)