Abstract
A one parameter family of algorithms is studied, which contains both the arithmetic-geometric mean of Gauss and its generalization by Borchardt, recently studied by J. and P. Borwein. We prove that the presence of an asymptotic formula for such an algorithm is, in view of the Poisson summation formula, equivalent to the vanishing of certain integrals. In the case of Gauss and Borchardt the latter involve theta functions. Finally, we investigate the question of convergence of the algorithm for complex values, thereby generalizing the corresponding result of Gauss.
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