International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 2, Pages 235-245
doi:10.1155/S016117128900027X
Generalizing the arithmatic geometric mean — a hapless computer experiment
Matematiska institutionen, Box 6701, Stockholm S-113 85, Sweden
Received 1 May 1988; Revised 20 September 1988
Copyright © 1989 Jaak Peetre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ”hapless computer experiment” in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ”fluctuations” are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt. The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ”hapless computer experiment” in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ”fluctuations” are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt.