An asymptotic relation for the zeros of Bessel functions

Abstract

We define the function j νκ for all real κ > 0 as follows: for κ = 1, 2, … the j νκ denotes the kth positive zero of the Bessel function J ν ( z) of first kind and for k − 1 < κ < k, j νκ denotes the kth positive zero of the cylinder Bessel function C ν ( z) = cos α J ν ( z) − sin α Y ν ( Z) with α = ( k − ν) π (see [2]), where Y ν ( x) is the Bessel function of second kind. We introduce the function ι( x) for x > − 1, l(x)= lim κ→∞ j κ,x,κ κ . and we prove, among other things, the inequality j νκ < κι( ν κ ) . Moreover, we find the first three terms of the asymptotic expansion of ι( x), for large values of x and other properties of this function.