A \(q\)-logarithmic analogue of Euler's sine integral
type: Item
instance of: scholarly article
title: A \(q\)-logarithmic analogue of Euler's sine integral
zbMATH Open document ID: 1118.11038
author: Nobushige Kurokawa
author: Masato Wakayama
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2007-04-26T00:00:00Z
full work available at URL: https://eudml.org/doc/108668
review text: Euler calculated the integral \[ \int_0^{\frac\pi2} \log(\sin x)\,dx=-\frac\pi2\log2. \] Let \(q>1\) and let \[ \ell_q(x)=(q-1)\sum_{m=1}^\infty\frac{x-1}{x-1+q^{m}} \] be a \(q\)-analogue of the usual logarithm. For \(0<a\leq1\) the authors consider generalizations \[ \int_0^{\frac\pi2} \log(1-ae^{2ix})\,dx\quad \text{and}\quad\int_0^{\frac\pi2} \ell_q(1-ae^{2ix})\,dx. \] The first integral is evaluated by Euler's dilogarithm function, the second is evaluated by the quantum dilogarithm function. In the limit case, when \(q\) tends to 1, both integrals coincide.
reviewed by: Ramūnas Garunkštis
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 5147109
MaRDI profile type: Publication
cites work: Q4335013
cites work: Multiple sine functions
cites work: Q4544426
cites work: Q5693285
cites work: Q4138810
cites work: Sur une formule classique
description: scientific article; zbMATH DE number 5147109