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, who introduced them in 1890.

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

cm 3 ⁡ ( x ) + sm 3 ⁡ ( x ) = 1 sm ⁡ ( π 3 3 − z ) = cm ⁡ ( z ) , where π 3 = B ( 3 , 3 ) and B is the Beta function sm ⁡ ( z exp ⁡ ( 2 i π 3 ) ) = exp ⁡ ( 2 i π 3 ) sm ⁡ ( z ) cm ⁡ ( z exp ⁡ ( 2 i π 3 ) ) = cm ⁡ ( z ) sm ′ ⁡ ( z ) = cm 2 ⁡ ( z ) cm ′ ⁡ ( z ) = − sm 2 ⁡ ( z ) sm ⁡ ( z ) = 6 ℘ ( z ; 0 , 1 27 ) 1 − 3 ℘ ′ ( z ; 0 , 1 27 ) cm ⁡ ( z ) = 3 ℘ ′ ( z ; 0 , 1 27 ) + 1 3 ℘ ′ ( z ; 0 , 1 27 ) − 1 where ℘ is Weierstrass's elliptic function