Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function $J_{\nu }(x)$

Abstract

A recent asymptotic expansion for the positive zeros $x=j_{\nu ,m}$ ($m=1,2,3,\dots$) of the Bessel function of the first kind $J_{\nu }(x)$ is studied, where the order $\nu$ is positive. Unlike previous well-known expansions in the literature, this is uniformly valid for one or both $m$ and $\nu$ unbounded, namely $m=1,2,3,\dots$ and $1\le \nu <\infty$. Explicit and simple lower and upper error bounds are derived for the difference between $j_{\nu ,m}$ and the first three terms of the expansion. The bounds are sharp in the sense they are close to the value of the fourth term of the expansion (i.e. the first neglected term).