Calculation of the complex zeros of the modified bessel function of the second kind and its derivatives

Abstract

A new algorithm is given for calculating with high accuracy all the complex zeros of the modified Bessel function of the 2nd kind and its derivatives, when the argument takes complex values, and the index takes both real and complex values. The algorithm is based on new asymptotic formulate for the modified functions of the 1st and 2nd kind, and on a Newton iterative scheme. Curves show how the zeros move as the index varies, and there are tables of the zeros, calculated to nine significant digits.