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

TL;DR

This paper derives explicit error bounds for a uniform asymptotic approximation of the positive zeros of the Bessel function $J_{ u}(x)$, valid for large and unbounded order and zero index, improving understanding of their accuracy.

Contribution

It introduces a uniform asymptotic expansion for Bessel zeros with explicit, sharp error bounds applicable for unbounded order and zero index.

Findings

Derived explicit lower and upper error bounds for the zeros

Bounds are close to the first neglected term, indicating sharpness

Expansion is valid for unbounded order and zero index

Abstract

A recent asymptotic expansion for the positive zeros $x=j_{\nu,m}$ ($m=1,2,3,\ldots$) 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 $m=1,2,3,\ldots$ and $1 \leq \nu < \infty$. 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).