Abstract
In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15.
Highlights
The elliptic integrals were first classified by Euler and Legendre, and Gauss, Jacobi and Abel started to study their inverses, the elliptic functions
In 1829, had found quickly converging Fourier series expansions for most of the twelve elliptic functions, which have been put in q-hypergeometric form in the authors article [1]
There are many series expansions for squareroots of rational functions of elliptic functions; as a bonus we prove some of these
Summary
The elliptic integrals were first classified by Euler and Legendre, and Gauss, Jacobi and Abel started to study their inverses, the elliptic functions. In 1829, had found quickly converging Fourier series expansions for most of the twelve elliptic functions, which have been put in q-hypergeometric form in the authors article [1]. As. Gudermann [2] showed, there are second series expansions for the twelve elliptic functions, starting from the imaginary period, which are not so quickly converging for all values of the variables; these expansions were found, without proof, by Glaisher [3]. Gudermann [2] showed, there are second series expansions for the twelve elliptic functions, starting from the imaginary period, which are not so quickly converging for all values of the variables; these expansions were found, without proof, by Glaisher [3] According to Heine, these 12 series can be written as follows
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