Computation of elliptic functions
Charles L. Critchfield · Journal of Mathematical Physics · 1989
The duplication formula for Weierstrass’ elliptic function is the basis for a simple algorithm by which one can calculate values of that function. As in the case for sin θ, this method is more efficient when the argument is reduced to a value near the origin by addition or subtraction of a suitable number of full periods of the function. When the invariants g2 and g3 are real numbers, (z;g2,g3) is periodic on both the real and imaginary axes of z, although these periods are not necessarily the basic ones. Convenient formulas for determining the relevant periods are given. They include an expression for q(k2) in terms of Gauss’ arithmetico-geometrical sequence. The method is applied to Jacobian functions by finding the invariants that represent a given k2. When this parameter is real the periods are well known and the Jacobian functions are calculated directly without transforming to those for which the parameter lies between zero and unity.