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 hexagons. 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:
- [math]\displaystyle{ \operatorname{cm}^3(x) + \operatorname{sm}^3(x) = 1 }[/math]
- [math]\displaystyle{ \operatorname{sm}\left( \frac{\pi_3} 3 - z \right) = \operatorname{cm}(z), }[/math] where [math]\displaystyle{ \pi_3 = B\left( \frac 1 3, \frac 1 3\right) }[/math] and [math]\displaystyle{ B }[/math] is the Beta function
- [math]\displaystyle{ \operatorname{sm}\left( z \exp\left( \frac{2i\pi} 3\right) \right) = \exp\left( \frac{2i\pi} 3 \right) \operatorname{sm}(z) }[/math]
- [math]\displaystyle{ \operatorname{cm} \left( z \exp\left( \frac{2i\pi} 3 \right) \right) = \operatorname{cm}(z) }[/math]
- [math]\displaystyle{ \operatorname{sm}'(z) = \operatorname{cm}^2(z) }[/math]
- [math]\displaystyle{ \operatorname{cm}'(z) = -\operatorname{sm}^2(z) }[/math]
- [math]\displaystyle{ \operatorname{sm}(z) = \frac{6\wp\left( z; 0, \frac 1 {27} \right)}{1 - 3\wp'\left(z;0,\frac 1 {27} \right)} }[/math]
- [math]\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} }[/math] where [math]\displaystyle{ \wp }[/math] is Weierstrass's elliptic function
See also
- Abel elliptic functions
- Jacobi elliptic functions
- Lee conformal world in a tetrahedron
- Weierstrass elliptic functions
Notes and references
- ↑ van Fossen Conrad, Eric (July 2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". Séminaire Lotharingien de Combinatoire 54: Art. B54g, 44. Bibcode: 2005math......7268V.
- ↑ Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve x3 + y3 - 3αxy = 1". Quarterly Journal of Pure and Applied Mathematics XXIV: 167–233. https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={%22pages%22:%5b179%5d}.
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