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 , and , determined by the algorithm
. Note that a 1 and b 1 are the respective arithmetic and geometric means of a and b, a 2 and b 2 the corresponding means of a 1 and b 1, etc. Thus the limit
really does deserve to be called the arithmetic-geometric mean of a and b. This algorithm first appeared in a paper of Lagrange, but it was Gauss who really discovered the amazing depth of this subject. Unfortunately, Gauss published little on the agM (his abbreviation for the arithmetic-geometric mean) during his lifetime. It was only with the publication of his collected works [12] between 1868 and 1927 that the full extent of his work became apparent. Immediately after the last volume appeared, several papers (see [15] and [35]) were written to bring this material to a wider mathematical audience. Since then, little has been done, and only the more elementary properties of the agM are widely known today.


Modular Form Elliptic Function Theta Function Fundamental Domain Modular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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-HeidelbergNew York, 1981.MATHGoogle Scholar
  5. [5]
    Carlson, B. C. Algorithms involving Arithmetic and Geometric Means. Amer. Math. Monthly 78 (1971), 496–505.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Cassels, J. W. S. Rational Quadratic Forms. Academic Press, Ncw York, 1978.MATHGoogle Scholar
  7. [7]
    Copson, E. T. An Introduction to the Theory of Functions r f 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]
    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.Google Scholar
  14. [14]
    Wie Gauss zur elliptischen Modulfunktion kam. Deutsche Mathematik 5 (1940), 158–175.Google Scholar
  15. [15]
    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. I. Wiley, New York, 1910.MATHGoogle 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 et Intégrals Abéliennes. In Ahrégé 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.MATHGoogle Scholar
  21. [21]
    Kline, M. Mathematical Thought from Ancient to Modern Times. Oxford U. Press, New York, 1972.Google Scholar
  22. [22]
    Lagrange, J. L. Oeuvres, Vol. I1. 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]
    Miei., G. Of Calculations Past and Present: The Archimedean Algorithm. Amer. Math. Monthly 90 (1983), 17–35.MathSciNetCrossRefGoogle Scholar
  27. [27]
    Mumford, D. Tata Lectures on Theta I. Progress in Mathematics Vol. 28, Birkhäuser, Boston, 1983.Google Scholar
  28. [28]
    Rosen, M. Abel’s Theorem on the Lemniscate. Amer. Math. Monthly 88 (1981), 387–395.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Serre, J.-P. Cours d’Arithmétique. Presses U. de France, Paris, 1970.Google Scholar
  30. [30]
    Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton U. Press, Princeton, 1971.Google 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.MATHGoogle Scholar
  33. [33]
    Todd, J. The Lemniscate Constants. Comm. of the ACM 18 (1975), 14–19.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Pot, B. Démonstration Elémentaire de la Relation 01 = 0;1 + 01 entre les Différentes Fonctions de Jacobi. L’Enseignement Math. 1 (1955), 258–261.Google Scholar
  35. [35]
    David, L. Arithmetisch-gcometrisches Mittel und Modulfunktion. J. für die Reine u. Ang. Math. 159 (1928), 154–170.MATHGoogle Scholar
  36. [36]
    Whittaker, E. T. and G. N. Watson. A Course of Modern Analysis, 4th cd. Cambridge U. Press, Cambridge, 1963.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

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

Personalised recommendations