# 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, $\ldots$.