Ratios of Bessel functions and roots of \(\alpha J_v(x)+xJ_v'(x)=0\)
type: Item
instance of: scholarly article
title: Ratios of Bessel functions and roots of \(\alpha J_v(x)+xJ_v'(x)=0\)
zbMATH Open document ID: 0938.33002
author: L. J. Landau
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-03T00:00:00Z
review text: The author shows that \(\zeta J_\nu(\nu \zeta)/J_{\nu+1} (\nu \zeta)\) is stricty decreasing in \(\nu\) on any interval not containing a singularity and obtaining an upper bound on its derivative: \[ {\partial \over \partial\nu} \left[{J_\nu (\nu\zeta)\over J_{\nu+1} (\nu\zeta)} \right]\leq-{2 \over \nu^2}. \] He also shows that \(x{\partial\over \partial v}[{J\nu(x) \over J_{\nu+1} (x)}]\geq 2\) where \(xJ_\nu(x)/J \nu+1(x)\) is strictly increasing in \(\nu\). The graph of \(J_\nu(x)/J_{\nu+1}(x)\) as a function of \(x\) is illustrated in figures for various values of \(\nu\). These results generalize and sharpen previously known results, and allow the deduction of a more complete description, including monotonicity and multiplicity, of the positive roots of the equation \(\alpha J\nu(x)+xJ_\nu'(x)=0\) for all real \(\nu\) and \(\alpha\).
reviewed by: R. S. Dahiya
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 1398604
zbMATH Keywords: Bessel functions
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6608
OpenAlex ID: W2014902043
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DOI: 10.1006/JMAA.1999.6608
description: scientific article; zbMATH DE number 1398604