Some Problems in the Precise Numeration of Fourier Expansions of Jacobian Elliptic Functions.

Abstract

Jacobian elliptic functions are useful in solving certain types of nonlinear problems, particularly in obtaining exact solutions of homogeneous Duffing equations. In addition to the theoretical formulation, the task of accurate numeration requires much effort. Finding the effective relationship inherent in the theory of elliptic functions may be very useful in developing algorithms/programs. This paper introduces some new formulations which assist the precise Fourier expansion of elliptic functions. The authors propose a set of optimal development neighborhoods of the respective values of modulus parameter m = 0, 1/2 and 1, which successfully renders uniform accuracy over the whole range of m. Also discussed are an exact solution of the snap-through spring Duffing free oscillator, an anomalous error propagation in the numerical transmission between words in the computer and the explicit polynomial solution of the so-called q-factor as a function of m.