Zeros of Bessel cross-products coming from oblique derivative boundary value problems

TL;DR

This paper investigates the zeros of Bessel cross-products from oblique derivative boundary problems in a circular annulus, revealing two types of zeros and providing asymptotic and numerical analyses of their behavior.

Contribution

It introduces a detailed analysis of zeros of Bessel cross-products arising from oblique boundary conditions, including asymptotic formulas and numerical plots.

Findings

Zeros include diverging and finite types.

Asymptotic expressions derived for fixed angles and thin annuli.

Numerical plots illustrate zero behavior with changing oblique angles.

Abstract

The paper is devoted to (combinations of) Bessel cross-products that arise from oblique derivative boundary value problems for the Laplacian in a circular annulus. We show that like their Neumann-Laplacian counterpart (and unlike the Dirichlet-Laplacian), they possess two kinds of zeros: those that can be derived by McMahon series and diverge to infinity in the limit, and exceptional ones that remain finite. For both cases we find asymptotic expressions for a fixed oblique angle and vanishing thickness of the annulus. We further present plots of numerically computed zeros and discuss their behaviour when the oblique angle changes and the thickness remains fixed.