On the $\nu$-zeros of the Bessel functions of purely imaginary order

TL;DR

This paper investigates the zeros of Bessel functions of purely imaginary order, providing asymptotic estimates for these zeros across different types of Bessel functions, expanding understanding of their zero distributions.

Contribution

It introduces a unified approach to asymptotically estimate the zeros of various Bessel functions of purely imaginary order, including $L_{i u}(x)$, $K_{i u}(x)$, and $J_{i u}(x)$, extending prior known results.

Findings

Asymptotic estimates for $ u$-zeros of $L_{i u}(x)$, $K_{i u}(x)$, and $J_{i u}(x)$ are derived.

The zeros of $K_{i u}(x)$ follow $ u_n o rac{ ext{constant} imes n}{ ext{log} ext{n}}$ as $n o fty$.

Unified asymptotic formulas are established for different Bessel functions of purely imaginary order.

Abstract

The $\nu$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{i\nu}(x)$, it is known that it possesses a countably infinite sequence of real $\nu$-zeros described by $\nu_n\sim \pi n/\log\,n$ as $n\to\infty$. Here we apply a unified approach to determine asymptotic estimates of the $\nu$-zeros of the modified Bessel functions $L_{i\nu}(x)\equiv I_{i\nu}(x)+I_{-i\nu}(x)$ and $K_{i\nu}(x)$ and the ordinary Bessel functions $J_{i\nu}(x)\pm J_{-i\nu}(x)$.