Abel elliptic functions
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Abel elliptic functions are holomorphic functions of one complex variable and with two periods. They were first established by Niels Henrik Abel and are a generalization of trigonometic functions. Since they are based on elliptic integrals, they were the first examples of elliptic functions. Similar functions were shortly thereafter defined by Carl Gustav Jacobi. In spite of the Abel functions having several theoretical advantages, the Jacobi elliptic functions have become the standard. This can have to do with the fact that Abel died only two years after he presented them while Jacobi could continue with his exploration of them throughout his lifetime. Both the elliptic functions of Abel and of Jacobi can be derived from a more general formulation which was later given by Karl Weierstrass based on their double periodicity.
History
The first elliptic functions were found by Carl Friedrich Gauss around 1795 in connection with his calculation of the lemniscate arc length, but first published after his death.[1] These are special cases of the general, elliptic functions which were first investigated by Abel in 1823 when he still was a student.[2] His starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. The year after Abel could report that his new functions had two periods.[3] Especially this property made them more interesting than the normal trigonometric functions which have only one period. In particular it meant that they had to be 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.[4] At the end of the same year he became aware of Carl Gustav Jacobi 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 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 .
References
- ^ J. Stillwell, Mathematics and Its History, Springer, New York (2010). ISBN 978-1441960528.
- ^ A. Stubhaug, Niels Henrik Abel and his Times, Springer-Verlag, Berlin (2000). ISBN 3-540-66834-9.
- ^ Oystein Ore, Niels Henrik Abel – Mathematician Extraordinary, AMS Chelsea Publishing, Providence, RI (2008). ISBN 978-0821846445.
- ^ N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 2, 101–181 (1827).
Literature
- C. Houzel, The Work of Niels Henrik Abel, in O.A. Laudal and R. Piene, The Legacy of Niels Henrik Abel – The Abel Bicentennial, Oslo 2002, Springer Verlag, Berlin (2004). ISBN 3-540-43826-2.