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An Extended Table of Zeros of Cross Products of Bessel Functions

Report Number: ARL 66-0023
Author(s): Henry E. Fettis; James C. Caslin
Corporate Author: Applied Mathematics Research Laboratory
Laboratory: Aerospace Research Laboratories
Date of Publication: 1966-02
Pages: 126
Contract: Laboratory Research - No Contract
Project: 7071
AD Number: AD0637474

Abstract:

The report contains tables of the first five roots of the following transcendental equations:

(a) J0(α) Y0(kα) = Y0(α) J0(kα)
(b) J1(α) Y1(kα) = Y1(α) J1(kα)
(c) J0(α) Y1(kα) = Y0(α) J1(kα)

where J0(α), Y0(α), J1(α), Y1(α) are Bessel functions of order 0 and 1 respectively. In these equations, α is the unknown and k is a parameter which may assume any positive value, other than 0 or 1. However, because of symmetry, it is sufficient in the first two cases to tabulate the roots only for 0 < k < 1. Following a suggestion of Bogert [1]*, additional tables are included listing an auxiliary quantity γ which is better suited to interpolation particularly when k is close to unity. The auxiliary function is defined in the three cases as follows:

cases (a) and (b): γn=((1-k)/nπ)αn;

case (c): γn=(|k-1|)/(n-1/2)π)αn,

where αn (n= 1,2,3...) is any root of the equation applicable to the case considered. The function γn has the important property that

Limγn=1
k->1


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