Dixon elliptic functions

In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity , as real functions they parametrize the cubic Fermat curve , just as the trigonometric functions sine and cosine parametrize the unit circle . They were named sm and cm by Alfred Dixon in 1890, by analogy to the trigonometric functions sine and cosine and the Jacobi elliptic functions sn and cn; Göran Dillner described them earlier in 1873. The functions sm and cm can be defined as the solutions to the initial value problem: Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral: which can also be expressed using the hypergeometric function: Both sm and cm have a period along the real axis of with the beta function and the gamma function: They satisfy the identity . The parametric function parametrizes the cubic Fermat curve with representing the signed area lying between the segment from the origin to , the segment from the origin to , and the Fermat curve, analogous to the relationship between the argument of the trigonometric functions and the area of a sector of the unit circle. To see why, apply Green's theorem: Notice that the area between the and can be broken into three pieces, each of area : The function has zeros at the complex-valued points for any integers and , where is a cube root of unity, (that is, is an Eisenstein integer). The function has zeros at the complex-valued points . Both functions have poles at the complex-valued points . On the real line, , which is analogous to .