In previous papers of the authors [Z. Angew. Math. Mech. 76, No. 7, 419-422 (1996;

Zbl 0878.33001)] the localization and computation of real zeros was treated, in this paper the notion of the topological degree was used to compute the total number

$N$ of complex zeros of a Bessel function within an open and bounded region of the complex plane. The localization of these zeros can also be computed. It was shown that the topological degree provides the total number of the complex roots. These roots are calculated using a generalized bisection method and the characteristic polyhedron criterion. The methods used require only the algebraic signs of the real and imaginary parts of the functions. An example is shown representing the

$N$ of various Bessel functions within several regions.