On the $\nu$-zeros of the modified Bessel function $K_{i\nu}(x)$ of positive argument

TL;DR

This paper refines the asymptotic estimates for the zeros of the modified Bessel function $K_{i u}(x)$ of imaginary order, providing more precise behavior for large zero indices and supporting findings with numerical validation.

Contribution

It offers a more accurate asymptotic estimate for the zeros of $K_{i u}(x)$, improving upon previous approximations using known asymptotic expansions.

Findings

Derived a refined asymptotic formula for zeros of $K_{i u}(x)$

Numerical results confirm the accuracy of the new estimates

Enhanced understanding of the zero distribution for large $n$

Abstract

The modified Bessel function of the second kind $K_{i\nu}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $\nu_n\sim \pi n/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimate for the bahaviour of these zeros for large $n$ by making use of the known asymptotic expansion of $K_{i\nu}(x)$ for large $\nu$. Numerical results are presented to illustrate the accuracy of the expansion obtained.