## Arithmetic-Geometric Mean

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)

so
 (6)

The AGM is very useful in computing the values of complete Elliptic Integrals and can also be used for finding the Inverse Tangent. The special value is called Gauss's Constant.

The AGM has the properties

 (7)

 (8)

 (9)

 (10)

The Legendre form is given by
 (11)

where and
 (12)

Solutions to the differential equation
 (13)

are given by and .

A generalization of the Arithmetic-Geometric Mean is

 (14)

which is related to solutions of the differential equation
 (15)

When or , there is a modular transformation for the solutions of (15) that are bounded as . Letting be one of these solutions, the transformation takes the form
 (16)

where
 (17) (18)

and
 (19)

The case gives the Arithmetic-Geometric Mean, and gives a cubic relative discussed by Borwein and Borwein (1990, 1991) and Borwein (1996) in which, for and defined by
 (20)

 (21)

For iteration with and and
 (22) (23)

 (24)

Modular transformations are known when and , but they do not give identities for (Borwein 1996).

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.