Zeros of Bessel Functions and Eigenvalues of Non-self-adjoint Boundary Value Problems

Abstract

For many non-self-adjoint boundary value problems involving the wave equation and the reduced wave equation in domains exterior to cylinders and spheres, it has recently been shown [1]-[5] that new representations of the solutions may be found systematically. In modern applications (e.g., potential scattering and diffraction phenomena) those alternative representations are usually more rapidly convergent than the classical ones, and asymptotic expansions with respect to parameters can often be found from them. These investigations require extensive knowledge of the zeros of certain linear combinations of the Hankel function and its first derivative considered as functions of the order of the Hankel function (with fixed argument). The most complete studies of these zeros are those of Magnus and Kotin [6] and Keller, Rubinow, and Goldstein [7]. The latter paper stressed their importance as poles of scattering amplitudes.