An approximation for the zeros of Bessel functions

Summary LetC_vk be thekth positive zero of the cylinder functionC_v(x)=cosJ_v(x)–sinY_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. 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.Dedicated to the memory of Professor Lothar CollatzThis work has been supported by the Hungarian Scientific Grant No. 6032/6319