The arithmetic-geometric mean (AGM) of two numbers and is defined by starting with and
, then iterating

(1) | |||

(2) |

until . and converge towards each other since

(3) |

But , so

(4) |

(5) |

(6) |

The AGM has the properties

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

A generalization of the Arithmetic-Geometric Mean is

(14) |

(15) |

(16) |

(17) | |||

(18) |

and

(19) |

(20) |

(21) |

(22) | |||

(23) |

(24) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``The Process of the Arithmetic-Geometric Mean.'' §17.6 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 571 ad 598-599, 1972.

Borwein, J. M. Problem 10281. ``A Cubic Relative of the AGM.'' *Amer. Math. Monthly* **103**, 181-183, 1996.

Borwein, J. M. and Borwein, P. B. ``A Remarkable Cubic Iteration.''
In *Computational Method & Function Theory: Proc. Conference Held in Valparaiso, Chile, March 13-18, 1989*
(Ed. A. Dold, B. Eckmann, F. Takens, E. B Saff, S. Ruscheweyh, L. C. Salinas, L. C., and R. S. Varga). New York: Springer-Verlag, 1990.

Borwein, J. M. and Borwein, P. B. ``A Cubic Counterpart of Jacobi's Identity and the AGM.''
*Trans. Amer. Math. Soc.* **323**, 691-701, 1991.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England: Cambridge
University Press, pp. 906-907, 1992.

© 1996-9

1999-05-25