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Axioms 2015, 4(3), 235-253; doi:10.3390/axioms4030235

On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions

Department of Mathematics, Uppsala University, P.O. Box 480, Uppsala SE-751 06, Sweden
Academic Editor: Hans J. Haubold
Received: 18 May 2015 / Revised: 10 June 2015 / Accepted: 19 June 2015 / Published: 8 July 2015
View Full-Text   |   Download PDF [235 KB, uploaded 8 July 2015]

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. View Full-Text
Keywords: hyperbolic series expansion; Carlson’s modern notation; hyperbolic modular function; Möbius transformation; Schwarz-Christoffel transformation; Peeta function hyperbolic series expansion; Carlson’s modern notation; hyperbolic modular function; Möbius transformation; Schwarz-Christoffel transformation; Peeta function
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Ernst, T. On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions. Axioms 2015, 4, 235-253.

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