The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice ?; in essence, the Weierstrass functions are naturally defined on a torus " T " = "'C "'/ ?.
12.
By starting with the Weierstrass p-function and associating with it a group of doubly periodic functions with two simple poles, he was able to give a simple derivation of the Jacobian elliptic functions, as well as modifying the existing notation to provide a more systematic approach to the subject.
13.
To solve equations ( 1.1 ) and ( 1.2 ) for a free-standing, doubly periodic surface, we consider an infinite 2D periodic surface occupying the entire x-y plane, and assume a discrete plane wave expansion for all currents, fields and potentials ( Tsao [ 1982 ], Scott [ 1989 ], Fourier optics ):
14.
Given a doubly periodic metric on \ mathbb R ^ 2 ( e . g . an imbedding in \ mathbb R ^ 3 which is invariant by a \ mathbb Z ^ 2 isometric action ), there is a nonzero element g \ in \ mathbb Z ^ 2 and a point p \ in \ mathbb R ^ 2 such that \ operatorname { dist } ( p, g . p ) ^ 2 \ leq \ frac { 2 } { \ sqrt { 3 } } \ operatorname { area } ( F ), where F is a fundamental domain for the action, while \ operatorname { dist } is the Riemannian distance, namely least length of a path joining p and g . p.
