Numerical techniques for finding 𝜈-zeros of Hankel functions

Abstract

Abstract. This paper is concerned with numerical procedures for the evaluation of the zeros, with respect to order, of Hankel functions and their derivatives in cases when the arguments of these functions are held fixed. Using Olver's asymptotic expansions, two auxiliary tables have been computed, one appropriate for real and the other for purely imaginary argument. These tables, included herein, permit the calculation of rather accurate approximations to the desired v-zeros for wide ranges of argument and index. Moreover, from the given tabular entries, the errors attendant with any approximate v-zero so determined can be easily estimated.

Complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑍)

“…The methods developed by Olver, however, are not entirely applicable in the present case, since the convergence only improves with large orders of n. On the other hand, the rational approximations given by Luke have been proved, under appropriate restrictions of the parameters, to converge in the first quadrant; convergence for |arg Z\ < it has recently been established by Fields [3]. It should be observed that Cochran and Hoffspiegel [4] have extended the work of Olver for determining the positive real or purely imaginary zeros of Hankel functions with respect to noninteger orders when the variable is held fixed. However, their methods are inapplicable, since the zeros of the modified Bessel functions are seen to be always complex.…”

Zeros of the Macdonald function of complex order

“…For the location of these, the Newton method has been used. Let us mention, however, other existing techniques like, for instance, the auxiliary tables for the evaluation of the -zeros of K (z), for z real or pure imaginary, proposed by Cochran and Hoffspiegel [4], the finite element approximation method, applied by Leung and Ghaderpanah [22] to a very precise determination of the zeros of K n (z), and a procedure, applied by Segura [28] to the computation of zeros of Bessel and other special functions, that uses fixed point iterations and does not require the evaluation of the functions.…”

An Application of the Finite Element Approximation Method to Find the Complex Zeros of the Modified Bessel Function K n (z)

“…Olver [10] evaluated zeros of Bessel functions of large order using uniform asymptotic expansions. For the case of Hankel functions, this work has been extended by Cochran and Hoffspiegel [1]. Based on the McMahon and Olver expansions, Doring [2] derived a method for the evaluation of complex zeros of cylinder functions.…”