Calculation of the complex zeros of a Bessel function of the second kind and its derivatives

Abstract

New algorithms are proposed for calculating, with high accuracy, a Bessel function of the second kind Y v ( z) and its derivatives, and also their complex zeros, when v takes real values, and z takes complex values The algorithms are based on the application of asymptotic methods to calculate the initial approximations of the zeros and on their further refinement using Newton's iteration scheme. A thorough qualitative investigation of the complex zeros is carried out, and new relationships in their behaviour are revealed. A large number of complex zeros, some of which are presented in the form of tables, is calculated.