1989/11/01 by K. Sala · 1 citation
Social Sciences · Mathematics · #Religion and Sociopolitical Dynamics in Nigeria #Mathematics #Jacobian matrix and determinant #Elliptic function #Mathematical analysis #Amplitude #Elliptic integral #Function (biology) #Transformation (genetics) #Elliptic curve #Complex plane #Applied mathematics
paper · doi:10.1137/0520100
openalex publication_date 1989/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
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.