Elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions.
Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse.
Further development of this theory led to hyperelliptic functions and modular forms.
The abelian group is called the period lattice.
Geometrically the complex plane is tiled with parallelograms.
This quotient group, called an elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus.
[2] 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.
A non-constant elliptic function takes on every value the same number of times in
it is defined by It is constructed in such a way that it has a pole of order two at every lattice point.
[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
.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.
[9] Additionally he defined the functions[10] and After continuation to the complex plane they turned out to be doubly periodic and are known as Abel elliptic functions.
Jacobi considered the integral function 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] 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.
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.
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.
[20] Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.
