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arXiv · 2108.01447

On the $ν$-zeros of the modified Bessel function $K_{iν}(x)$ of positive argument

R B Paris
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Abstract

The modified Bessel function of the second kind $K_{iν}(x)$ 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 $ν_n\sim πn/\log\,n$ 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 $K_{iν}(x)$ for large $ν$. Numerical results are presented to illustrate the accuracy of the expansion obtained.

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R B Paris. 2022-04-07. On the $ν$-zeros of the modified Bessel function $K_{iν}(x)$ of positive argument. https://arxiv.org/abs/2108.01447

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