Calculation of the complex zeros of Hankel functions and their derivatives

Abstract

New algorithms are proposed for calculating with high accuracy the complex zeros of Hankel functions of the first and second kind when the index acquires real values and the argument acquires complex values. The algorithms are based on reducing the problem of calculating the zeros of Hankel functions to the problem — partly considered earlier by the authors — of calculating the zeros of a modified Bessel function of the second kind. Using asymptotic methods, analytical formulas are obtained which enable us to localize the zeros in the complex plane and find their first approximations, which are further refined using Newton's iteration scheme. A careful qualitative analysis of the zeros is made, programs∗∗ (∗∗One can consult the texts of all programs mentioned in this paper at the Computational Centre, Academy of Sciences of the USSR, Moscow.) for calculating both the Hankel functions and their derivatives themselves, and their zeros, are compiled, and a large number of zeros, some of which are presented in the paper together with graphs illustrating the regularity of the distribution of the zeros, are calculated.