In this paper the problem of computing the numerical value of the integral § q(z)/p(z)dz, where q and p are polynomials, given by their coefficients, and Г is a curve in the complex plane, is investigated from the point of view of (serial) bit complexity, i.e., finite precision arithmetic is used. The first algorithm presented computes this integral in the special case that the zeros of p lie in a small circle not intersected by Г. The second algorithm discussed in this paper computes a special type of partial fraction decomposition especially well suited for this application, but also of interest by itself. Combining these algorithms yields an algorithm for the computation of contour integrals of rational functions in the general case. The running time of the algorithms is estimated in terms of the error bound prescribed for the result, the degree of the polynomials involved, and the condition of the problem, measured by a lower bound for the distance between the zeros of p and the points of Г.

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