Evaluation of the Zeros of Cross-Product Bessel Functions

Abstract

227 be large, however, it appeared appropriate to make an independent investigation of the initial roots of (la) and (lb) by study of characteristic solutions of Bessel's equation in the interval a 5¡ r ^ b without explicit reference to the usual Bessel and Neumann functions. Approximate analytic formulas have been obtained from which estimates may be made of the characteristic values, for the case of the first Dirichlet root and for the first two roots subject to the Neumann boundary condition , and an independent numerical determination of the characteristic values and characteristic functions has been made with the CYCLONE electronic digital computer at Iowa State University for cases in which (6-a)/(b + a) was given the values 0.001, 0.01, and 0.1. It is the purpose of the present note to summarize the results of this investigation, for which more detailed results will be available elsewhere (see Section 5). 2. Transformation of Bessel's Equation. It may be noted that, due to the nature of the customary Bessel functions of high order, and in particular because the function Jn remains quite small until its argument is comparable to its order, the lowest characteristic values, q, will be in the neighborhood of n/b for n large. For this reason, and to focus attention on the interval a ^ r ^ b, it is convenient to define /" \ b-a (2a) v = b + a' (2b) i = ,«[(8*+J!y-»«], r-(b + o)/2 and (2c) x-2 b-a In terms of these quantities, (3) r = ^±-? (1 + vx), with-láiál, and Bessel's equation assumes the form The solutions to (4) which are of interest are those for which the Dirichlet boundary condition (Z = 0) or, alternatively, the Neumann boundary condition (dZ/dx = 0) applies at x = ±1. When the Dirichlet boundary condition is applied, it may be convenient for some purposes to make the transformation (5) S-(1 + vx)mZ, in terms of which (4) may be written withS(±l) = 0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use