The topological degree theory for the localization and computation of complex zeros of bessel functions

We study the complex zeros of Bessel functions of real order of the first and second kind and their first derivatives. The notion of the topological degree is employed for the calculation of the exact number of these zeros within an open and bounded region of the complex plane, as well as for localization of these zeros. First, we prove that the value of the topological degree provides the total number of complex roots within this region. Subsequently, these roots are computed by a generalized bisection method. The method presented here computes complex zeros of Bessel functions, requiring only the algebraic signs of the real and imaginary part of these functions. It has been implemented and tested, and performance results are presented.

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