“Arithmetic-Geometric Mean”的概念、定义、习题解题方法及翻译-数学辞典

单词

Arithmetic-Geometric Mean

释义

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)

Now, add

to each side

(5)

(6)

The AGM is very useful in computing the values of complete Elliptic Integrals and can also beused 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

, 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 andBorwein (1990, 1991) and Borwein (1996) in which, for

and

defined by

(20)

(21)

For iteration with

and

			(22)
			(23)

(24)

Modular transformations are known when

and

, but they do not give identities for

(Borwein 1996).

See also Arithmetic-Harmonic Mean

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.

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