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 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.