Royal Society Publishing

Asymptotic Behaviour of the Inflection Points of Bessel Functions

R. Wong, T. Lang

Abstract

Asymptotic expansions are derived for the inflection points j$_{\nu k}^{\prime \prime}$ of the Bessel function J$_{\nu}$(x), as k $\rightarrow \infty $ for fixed $\nu $ and as $\nu \rightarrow \infty $ for fixed k. Also derived is an asymptotic expansion of J$_{\nu}$(j$_{\nu k}^{\prime \prime}$) as $\nu \rightarrow \infty $. Finally, we prove that j$_{\nu \lambda}^{\prime \prime}\geq \nu \surd $2 if $\lambda \geq $ (0.07041)$\nu $ + 0.25 and $\nu \geq $ 7, which implies by a recent result of Lorch & Szego that the sequence {$|$J$_{\nu}$(j$_{\nu k}^{\prime \prime}$)$|$} is decreasing, for k = $\lambda $, $\lambda $+1, $\lambda $+2, $\ldots $.

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