The Arithmetic-Geometric Mean of Gauss

  • David A. Cox


The arithmetic-geometric mean of two numbers a and b is defined to be the common limit of the two sequences {a n } n = 0 and {b n } n = 0 determinec by the algorithm
$$\begin{array}{l} {a_0} = a,\quad \quad \quad \quad \quad \quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_0} = b, \\ {a_{n + 1}} = \left( {{a_n} + {b_n}} \right)/2,\;\quad \quad {b_{n + 1}} = {\left( {{a_n}{b_n}} \right)^{1/2}},\quad n = 0,1,2, \ldots \\ \end{array}$$


Modular Form Elliptic Function Theta Function Fundamental Domain Modular Function 
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  1. [1]
    Alling, N. L. Real Elliptic Curves. North-Holland Mathematics Studies, Vol. 54, North-Holland, Amsterdam, 1981.Google Scholar
  2. [2]
    Bernoulli, Jacob. Opera, Vol. I. Geneva, 1744.Google Scholar
  3. [3]
    Bernoulli, Johann. Opera omnia, Vol. I. Lausanne, 1742.Google Scholar
  4. [4]
    Bühler, W. K. Gauss : A Biographical Study. Springer-Verlag, Berlin-Heidelberg-New York, 1981.CrossRefzbMATHGoogle Scholar
  5. [5]
    Carlson, B. C. Algorithms involving Arithmetic and Geometric Means. Amer. Math. Monthly 78(1971), 496–505.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Cassels, J. W. S. Rational Quadratic Forms. Academic Press, New York, 1978.zbMATHGoogle Scholar
  7. [7]
    Copson, E. T. An I ntroduct ion to the Theory of Functions of a Complex Variable. Oxford U. Press, London, 1935.Google Scholar
  8. [8]
    Enneper, A. Elliptische Functionen : Theorie und Geschichte. Halle, 1876.Google Scholar
  9. [9]
    Euler, L. Opera Omnia, Series Prima, Vol. XX and XXI. Teubner, Leipzig and Berlin, 1912–1913.Google Scholar
  10. [10]
    Fuchs, W. Das arithmetisch-geometrische Mittel in den Untersuchungen von Carl Friedrich Gauss. Gauss-Gesellschaft Göttingen, Mittelungen No. 9(1972), 14–38.Google Scholar
  11. [11]
    Gauss, C. F. Disquisitiones Arithmeticae. Translated by A. Clark, Yale U. Press, New Haven, 1965 (see also [12, I]).Google Scholar
  12. [12]
    Gauss, C. F. Werke. Göttingen-Leipzig, 1868–1927.Google Scholar
  13. [13]
    Geppert, H. Bestimmung der Anziehung eines elliptischen Ringes. Ostwald’s Klassiker, Vol. 225, Akademische Verlag, Leipzig, 1927.zbMATHGoogle Scholar
  14. [14]
    Gauss, C. F. Wie Gauss zur elliptischen Modulfunktion kam. Deutsche Mathematik 5 (1940), 158–175.Google Scholar
  15. [15]
    Gauss, C. F. Zur Theorie des arithmetisch-geometrischen Mittels. Math. Annalen 99 (1928), 162–180.CrossRefGoogle Scholar
  16. [16]
    Gradshteyn, I. S. and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press, New York, 1965.Google Scholar
  17. [17]
    Hancock, H. Lectures on the Theory of Elliptic Functions. Vol. 1. Wiley, New York, 1910.Google Scholar
  18. [18]
    Hoffman, J. E über Jakob Bernoullis Beiträge zur Infinitesimalmathematik. L’Enseignement Math. 2 (1956), 61–171.Google Scholar
  19. [19]
    Houzel, C. Fonctions Elliptiques ct Intćgrals Abéliennes. In Abrégé d’histoire des mathématiques 1700–1900, Vol. II. Ed. by J. Dieudonné, Hermann, Paris, 1978, 1–112.Google Scholar
  20. [20]
    Jacobi, C. C. J. Gesammelte Werke. G. Reimer, Berlin, 1881.zbMATHGoogle Scholar
  21. [21]
    Kline, M. Mathematical Thought from Ancient to Modern Times. Oxford U. Press, New York, 1972.zbMATHGoogle Scholar
  22. [22]
    Lagrange, J. L. Oeuvres, Vol. II. Gauthier-Villars, Paris, 1868.Google Scholar
  23. [23]
    Legendre, A. M. Traité des Fonctions Elliptiques. Paris, 1825–1828.Google Scholar
  24. [24]
    Lockwood, E. H. A Book of Curves. Cambridge U. Press, Cambridge, 1971.Google Scholar
  25. [25]
    Markushevitch, A. I. Die Arbeiten von C. F. Gauss über Funktionentheorie. In C. F. Gauss Gedenkband Anlässlich des 100. Todestages am 23. Februar 1955. Ed. by H. Reichart, Teubner, Leipzig, 1957, 151–182.Google Scholar
  26. [26]
    Miel, G. Of Calculations Past and Present : The Archimedean Algorithm. Amer. Math. Monthly 90 (1983), 17–35.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    Mumford, D. Tata Lectures on Theta I. Progress in Mathematics Vol. 28, Birkhäuser, Boston, 1983.CrossRefzbMATHGoogle Scholar
  28. [28]
    Rosen, M. Abel’s Theorem on the Lemniscate. Amer. Math. Monthly 88 (1981), 387–395.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    Serre, J.-P. Cours d’Arithmétique. Presses U. de France, Paris, 1970.zbMATHGoogle Scholar
  30. [30]
    Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton U. Press, Princeton, 1971.zbMATHGoogle Scholar
  31. [31]
    Stirling, J., Methodus Differentialis. London, 1730.Google Scholar
  32. [32]
    Tannery, J. and J. Molk. Eléments de la Théorie des Fonctions Elliptiques, Vol. 2. Gauthiers-Villars, Paris, 1893.Google Scholar
  33. [33]
    Todd, J. The Lemniscate Constants. Comm. of the Acm 18 (1975), 14–19.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    van der Pol, B. Démonstration Elémentaire de la Relation Θ340424entre les Différentes Fonctions de Jacobi. L’Enseignement Math. 1 (1955), 258–261.Google Scholar
  35. [35]
    von David, L. Arithmetisch-geometrisches Mittel und Modulfunktion. J. für die Reine u. Ang. Math. 159 (1928), 154–170.zbMATHGoogle Scholar
  36. [36]
    Whittaker, E. T. and G. N. Watson. A Course of Modern Analysis, 4th cd. Cambridge U. Press, Cambridge, 1963.zbMATHGoogle Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • David A. Cox
    • 1
  1. 1.Department of MathematicsAmherst CollegeAmherstUSA

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