Dixon's elliptic functions

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In mathematics, Dixon's elliptic functions, are two doubly periodic meromorphic functions on the complex plane that have regular hexagons as repeating units: the plane can be tiled by regular hexagons in such a way that the restriction of the function to such a hexagon is simply a shift of its restriction to any of the other hexagons. This in no way contradicts the fact that a doubly periodic meromorphic function has a fundamental region that is a parallelogram: the vertices of such a parallelogram (indeed, in this case a rectangle) may be taken to be the centers of four suitably located hexgaons.

These functions are named after Alfred Cardew Dixon,[1] who introduced them in 1890.[2]

Dixon's elliptic functions are denoted sm and cm, and they satisfy the following identities:

where and is the Beta function
where is Weierstrass's elliptic function

See also

Notes and references

  1. ^ van Fossen Conrad, Eric; Flajolet, Philippe (July 2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". Séminaire Lotharingien de Combinatoire. 54: Art. B54g, 44. arXiv:math/0507268Freely accessible. Bibcode:2005math......7268V. MR 2223029. 
  2. ^ Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve x3 + y3 = 1". Quarterly Journal of Pure and Applied Mathematics. XXIV: 167–233. 
Retrieved from "https://en.wikipedia.org/w/index.php?title=Dixon%27s_elliptic_functions&oldid=846567020"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Dixon's_elliptic_functions
This page is based on the copyrighted Wikipedia article "Dixon's elliptic functions"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA