An approximation for the zeros of Bessel functions

Abstract

LetC vk be thekth positive zero of the cylinder functionC v(x)=cos?J v(x)?sin?Y v(x), whereJ v(x),Y v(x) are the Bessel functions of first kind and second kind, resp., andv>0, 0??<?. Definej vk byj vk=C vk with $$\kappa = k - \frac{\alpha }{\pi }$$ . Using the notation 1/K=?, we derive the first two terms of the asymptotic expansion ofj vk in terms of the powers of ? at the expense of solving a transcendental equation. Numerical examples are given to show the accuracy of this approximation.