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Unformatted text preview: Elliptic functions See [section 4.5] and  for more information. 1 Definition An elliptic function is a single-valued doubly-periodic function of a single complex variable which is analytic except at poles and whose only singularities in the finite plane are poles. Such functions are called elliptic because they define functions on the two-torus. Imagine building a two-torus (a doughnut) by taking a square and identifying opposing sides that means a function on the complex plane which is periodic in two directions can be thought of as a function on a square, periodic at opposing sides, and hence is a function on the two-torus. Given an angle , define u = integraldisplay d parenleftBig 1- m sin 2 parenrightBig 1 / 2 where m is the parameter , a real number in the interval 0 m 1. In terms of the elliptic integrals discussed in A-W section 5.8, u = F (sin | m ) where F is the elliptic integral of the first kind. The angle corresponding to u is known as the amplitude of u , and is denoted am u . Then, define the Jacobi elliptic functions sn u = sin cn u = cos dn u = parenleftBig 1- m sin 2 parenrightBig 1 / 2 In the case m = 0, note that sn...
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- Spring '11