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In this chapter we will consider the problem of computing all the zeros of an analytic function 1that lie in the interior of a Jordan curve"/. The algorithm that we will present computes not only approximations for the zeros but also their respective multiplicities. It doesn't require initial approximations for the zeros and gives accurate results. The algorithm is based on the theory of formal orthogonal polynomials. Its principal means of obtaining information about the location of the zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along"/. This form involves the logarithmic derivative I'/1 of 1. Our approach could therefore be called a logarithmic residue based quadrature method. In the next chapters we will see how it can be used to locate clusters of zeros of analytic functions, to compute all the zeros and poles of a meromorphic function that lie in the interior of a Jordan curve, and to …