In mathematics Abel elliptic functions are a special kind of
elliptic functions, that were established by the Norwegian mathematician
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
. He published his paper "Recherches sur les Fonctions elliptiques" in
Crelle's Journal in 1827. It was the first work on elliptic functions that was actually published.
Abel's work on elliptic functions also influenced
Jacobi's studies of elliptic functions, whose 1829 published book "
Fundamenta nova theoriae functionum ellipticarum" became the standard work on elliptic functions.
History
Abels starting point were the
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s which had been studied in great detail by
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
. He began his research in 1823 when he still was a student. In particular he viewed them as
complex functions which at that time were still in their infancy. In the following years Abel continued to explore these functions. He also tried to generalize them to functions with even more periods, but seemed to be in no hurry to publish his results.
But in the beginning of the year 1827 he wrote together his first, long presentation ''Recherches sur les fonctions elliptiques'' of his discoveries. At the end of the same year he became aware of
Carl Gustav Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occas ...
and his works on new transformations of elliptic integrals. Abel finishes then a second part of his article on elliptic functions and shows in an appendix how the transformation results of Jacobi would easily follow.
[ When he then sees the next publication by Jacobi where he makes use of elliptic functions to prove his results without referring to Abel, the Norwegian mathematician finds himself to be in a struggle with Jacobi over priority. He finishes several new articles about related issues, now for the first time dating them, but dies less than a year later in 1829.] In the meantime Jacobi completes his great work '' Fundamenta nova theoriae functionum ellipticarum'' on elliptic functions which appears the same year as a book. It ended up defining what would be the standard form of elliptic functions in the years that followed.
Derivation from elliptic Integrals
Consider the elliptic integral of the first kind in the following symmetric form:
: with .
is an odd increasing function on the interval