Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function $J_ν(x)$
T. M. Dunster
TL;DR
This work provides explicit, sharp error bounds for a uniform asymptotic expansion of the positive zeros $j_{\nu,m}$ of the Bessel function $J_{\nu}(x)$, which is valid uniformly for both the order $\nu$ and the index $m$. The zeros are approximated by $j_{\nu,m}/\nu \approx z_{m,0}+z_{m,1}/\nu^{2}+z_{m,2}/\nu^{4}$, and the paper derives two-sided bounds for the truncation error in terms of the next coefficient $z_{m,3}$ and a computable factor $\chi_m$, via a uniform Airy-based framework with the auxiliary function $\mathcal{Z}_3(\nu,z)=\zeta+\eta(\nu,z)$. The analysis combines an Airy-type error bound (Theorem Ai), a refinement that relates Airy-roots to the truncated expansion (Theorem e_bold), and auxiliary estimates (Hethcote, Qu and Wong) to yield the explicit constants in the bounds, demonstrating the bounds are sharp in that they are close to the size of the next neglected term. The results extend classical asymptotic expansions by achieving uniform validity in both $m$ and $\nu$, enabling reliable error control for computing Bessel zeros in regimes of large order or large zeros, beneficial for spectral problems and numerical applications.
Abstract
A recent asymptotic expansion for the positive zeros $x=j_{ν,m}$ ($m=1,2,3,\ldots$) of the Bessel function of the first kind $J_ν(x)$ is studied, where the order $ν$ is positive. Unlike previous well-known expansions in the literature, this is uniformly valid for one or both $m$ and $ν$ unbounded, namely $m=1,2,3,\ldots$ and $1 \leq ν< \infty$. Explicit and simple lower and upper error bounds are derived for the difference between $j_{ν,m}$ 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).