Dixon's elliptic functions

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:

${\displaystyle \operatorname {cm} ^{3}(x)+\operatorname {sm} ^{3}(x)=1}$

${\displaystyle \operatorname {sm} \left({\frac {\pi _{3}}{3}}-z\right)=\operatorname {cm} (z),}$ where ${\displaystyle \pi _{3}=B(3,3)}$ and $B$ is the Beta function

${\displaystyle \operatorname {sm} \left(z\exp \left({\frac {2i\pi }{3}}\right)\right)=\exp \left({\frac {2i\pi }{3}}\right)\operatorname {sm} (z)}$

${\displaystyle \operatorname {cm} \left(z\exp \left({\frac {2i\pi }{3}}\right)\right)=\operatorname {cm} (z)}$

${\displaystyle \operatorname {sm} '(z)=\operatorname {cm} ^{2}(z)}$

${\displaystyle \operatorname {cm} '(z)=-\operatorname {sm} ^{2}(z)}$

${\displaystyle \operatorname {sm} (z)={\frac {6\wp \left(z;0,{\frac {1}{27}}\right)}{1-3\wp '\left(z;0,{\frac {1}{27}}\right)}}}$

${\displaystyle \operatorname {cm} (z)={\frac {3\wp '\left(z;0,{\frac {1}{27}}\right)+1}{3\wp '\left(z;0,{\frac {1}{27}}\right)-1}}}$ where $\wp$ is Weierstrass's elliptic function

See also

• Abel elliptic functions
• Jacobi elliptic functions
• Weierstrass elliptic functions

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/0507268. 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: 167–245.