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 
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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

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

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