A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

Abstract

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.