Asymptotic relation for zeros of cross-product of Bessel functions and applications

Abstract

Let $a_{ν,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_ν(R z) Y_ν(z) - J_ν(z) Y_ν(R z)$, where $ν\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $α(x)$ characterizes the limit behavior of $a_{ν,k}$ in the following sense: $$ \lim_{k \to \infty} \frac{a_{kx,k}}{k} = α(x), \quad x \geq 0. $$ Moreover, we show that $$ a_{ν,k} < \frac{πk}{R-1} + \frac{πν}{2R}. $$ We use $α(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.