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One approach for computations with special functions in computer algebra is the systematic use of analytic functions whenever possible. This naturally leads to problems of how to answer questions about analytic functions in a fully effective way. Such questions comprise the determination of the radius of convergence or the evaluation of the analytic continuation of the function at the endpoint of a broken like path. In this paper, we propose a first definition for the notion of an effective analytic function and we show how to effectively solve several types of differential equations in this context. We will limit ourselves to functions in one variable.

**Keywords: **complex analysis, computer algebra,
algorithm, majorant method

**A.M.S. subject classification: **30-04, 34M99