Arithmetic-geometric mean
Abstract
The arithmetic-geometric mean M(a,b) is the common limit of sequences defined by recurrence relations. The function M (a, b) is a homogeneous infinitely multivalued function of its arguments. The properties of the function M (a, b) were specified by C. F. Gauss. However, the results of his research, published in posthumous editions, represent only separate notes, not connected by unity of presentation, and it is difficult to judge from them the form of a complete presentation of the subject. The latter has prompted numerous comments to reconstruct a coherent statement of the properties of the arithmetic-geometric mean, and a number of studies to improve or modify certain details of such a reconstruction. The properties of the function M (a, b) follow directly from the properties of the algorithm defining it. Meanwhile, the proposed statements of the properties of the arithmetic-geometric mean go beyond the defining algorithm and involve considerations that are superfluous to the task at hand.
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References
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