Computation of Elliptic Functions of Rational Fractions of a Quarterperiod

The evaluation of Jacobian elliptic functions with arguments which are rational fractions of a quarterperiod is often necessary in network design. Conventional methods, using theta functions, are quite satisfactory for this purpose, but, in the more interesting range of the modulus
k
near unity, they do not represent the best that can be done. A slight modification of the theta functions, together with a change of parameter made possible by the special nature of the rational-fraction form of argument, produces a computing scheme which, in most practical instances, converges more quickly than the classical one. The scheme is illustrated by a numerical example.