A very simple proof of the eta transformation formula
type: Item
instance of: scholarly article
title: A very simple proof of the eta transformation formula
zbMATH Open document ID: 1136.11032
author: Jürgen Elstrodt
published in: Manuscripta Mathematica
publication date: 2007-01-11T00:00:00Z
review text: The ``very simple proof'' of the title is based upon the behavior of the Weierstrass zeta function with respect to the lattice \(\mathbb Z+\mathbb Zr\) \((\Im\,\tau> 0)\); \[ \zeta(z)= {1\over z}+ \sum_{\substack{ m,n\in\mathbb Z\\ (m,n)\neq (0,0)}} \Biggl({1\over z+ m\tau+ n}- {1\over m\tau+ n}+ {z\over (m\tau+ n)}\Biggr), \] in particular, upon the behavior of \(\varphi_1(\tau)= {i\over 2\pi} \zeta({1\over 2})\) and \(\varphi_2(\tau)= {i\over 2\pi}\zeta(\tau/2)\) under the modular transformation \(\tau\to -1/\tau\). The proof applies Euler's partial fraction expansions of \(\pi\operatorname{cot}\pi z\) and \(\pi^2/(\sin^2\pi z)\). The author points out that his proof is a simplification of one suggested by \textit{H. Petersson} [Modulfunktionen und quadratische Formen, Berlin etc.: Springer-Verlag (1982; Zbl 0493.10033), Appendix D].
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 5115238
zbMATH Keywords: Dedekind eta function
zbMATH Keywords: Weierstrass zeta function
MaRDI profile type: Publication
full work available at URL: s00229-006-0048-0
OpenAlex ID: W2075761230
cites work: Q5605180
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cites work: Sur une formule classique
DOI: 10.1007/S00229-006-0048-0
description: scientific article; zbMATH DE number 5115238