Transformations of the Jacobian Amplitude Function and Its Calculation via the Arithmetic-Geometric Mean

Abstract

With the aid of the Poisson summation formula, expressions for the Jacobian amplitude function, ${\operatorname {am}}(z;m)$, along with the complete set of Jacobian elliptic functions are given that, aside from their branchpoints and poles, respectively, are convergent throughout the complex plane for arbitrary parameter m. By utilizing the expression for ${\operatorname {am}}(z;m)$, its periodicity properties are determined in each of the regions $m < 0$, $0 < m < 1$, and $m > 1$. Novel yet fundamental identities are presented describing various linear and quadratic transformations of the Jacobian amplitude function. Finally, that method based on the arithmetic-geometric mean and most widely employed for calculating the Jacobian elliptic functions is shown to be, when interpreted explicitly in terms of ${\operatorname {am}}(z;m)$ and its transformation properties, a method first and foremost for the calculation of the Jacobian amplitude and co-amplitude functions from which the elliptic functions themselves are subsequently evaluated by means of simple, trigonometric identities.