Abstract
The question of the convergence and the limits of several sequences similar to the classical arithmetic-geometric mean sequences of Gauss, but where arbitrary choices of the determination of the square roots involved are made, is examined.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. W. Borchardt, Gesammelte Werke, Reimer, Berlin, 1888.
B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), 696–705.
A. Erdélyi etcal., eds., Higher transcendental functions, II, McGraw-Hill, New York, 1953.
L. Fejér, Gesammelte Arbeiten, II, Birkhäser, Basel, 1970.
C. F. Gauss, Werke, I - XII, B. C. Teubner, Leipzig, 1870–1929.
H. Geppert, ed., Ostwalds Klassiker, #225, C. F. Gauss, Anziehung eines elliptischen Ringes, Akademische Verlagsgesellschaft, Leipzig, 1925.
H. Geppert, Zur Theorie des arithmetisch-geometrischen Mittels, Math. Ann. 99 (1928), 162–180.
H. Geppert, Wie Gauss zur elliptischen Modulfunktion kam, Deutsche Math. 5 (1940–41), 158–175.
Ahrlhawitz, Über die Einführung der elementaren transzendenten Funktionen in der algebraische Analysis, Math. Ann. 70 (1911), 33–47 Mathematische Werke, I, 706–721, Birkhäuser, Basel, 1932.
A. I. Markuschevich, Die Arbeiten von C. F. Gauss fiber Funktionentheorie, 151–182 in Reichardt [12].
A. I. Markuschevich, The remarkable sine functions, Elsevier, New York, 1966.
H. Reichardt, ed., Gauss 1777–1855, Gedenkband, B. G. Teubner, Leipzig, 1957.
John Todd, The lemniscate constants, Comm. ACM 18 (1975), 16–19.
John Todd, The many values of mixed means, II, to appear.
J. V. Uspensky, On the arithmetic-geometric means of Gauss, Math. Notae 5 (1945), 1488, 57–88, 129–161.
L. von Ddvid, Zur Gaussischen Theorie der Modulfunktion, Rend. Circ. Mat. Palermo, 35 (1913), 82–89.
L. von Ddvid, Arithmetisch-geometrisches Mittel und Modulfunktion, J. Reine Angew. Math. 159 (1928), 154–170.
E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge University Press, London, 1962.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1978 Springer Basel AG
About this chapter
Cite this chapter
Todd, J. (1978). The Many Limits of Mixed Means, I. In: Beckenbach, E.F. (eds) General Inequalities 1 / Allgemeine Ungleichungen 1. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 41. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5563-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5563-1_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5565-5
Online ISBN: 978-3-0348-5563-1
eBook Packages: Springer Book Archive