“Best Possible” Upper and Lower Bounds for the Zeros of the Bessel Function J_v(x)

Abstract:

Let j_v,k denote the k-th positive zero of the Bessel function j_v,k. In this paper, we prove that for í > 0 and k = 1, 2, 3, … ,

These bounds coincide with the first few terms of the well-known asymptotic expansion as v → ∞ k being fixed, where a_k is the k-th negative zero of the Airy function Ai(x), and so are “best possible”.