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

Abstract

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

16 pages, 2 figures. Remark 3.2 is added; some proofs are expanded and figures are improved according to referee's suggestions. Accepted to Journal of Mathematical Analysis and Applications