Cubic elliptic functions
type: Item
instance of: scholarly article
title: Cubic elliptic functions
zbMATH Open document ID: 1118.33009
author: Shaun Cooper
published in: The Ramanujan Journal
publication date: 2006-11-03T00:00:00Z
full work available at URL: http://hdl.handle.net/10179/4407
review text: The cubic elliptic functions referred to in the title of this paper are ones which are analogous to \[ g_1(\theta,q)= \frac 16 + \sum_{1\leq k < \infty} \frac{q^{k}}{1+q^{k}+q^{2k}} \sin(2k \theta) \] which is an elliptic function in \(\theta\) with fundamental periods \(2 \pi\) and \(6\pi \imath t\) where \(q=e^{-2\pi t}\). The development given here is in the style of Jacobi and Ramanujan. The fundamental formulae are proved either by algebraic transformations or by complex function theory. An analogue of a formula of Venkatachaliengar plays a particularly important role in the development. The author proves a large number of formulae relating different functions in the class considered.
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 5070742
zbMATH Keywords: Eisenstein series
zbMATH Keywords: elliptic function
zbMATH Keywords: modular transformation
zbMATH Keywords: Ramanujan
zbMATH Keywords: theta function
zbMATH Keywords: Venkatachaliengar
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
OpenAlex ID: W2144708940
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DOI: 10.1007/S11139-006-8481-Y
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description: scientific article; zbMATH DE number 5070742