GaussAGM - Maple Help

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GaussAGM

Gauss' arithmetic geometric mean

 Calling Sequence GaussAGM(a, b)

Parameters

 a - expression denoting a complex number b - expression denoting a complex number

Description

 • GaussAGM(a, b) is the limit of the iteration

${a}_{0}≔a$

${b}_{0}≔b$

${a}_{n+1}≔\frac{{a}_{n}}{2}+\frac{{b}_{n}}{2}$

${b}_{n+1}≔\left({a}_{n}+{b}_{n}\right)\sqrt{\frac{{a}_{n}{b}_{n}}{{\left({a}_{n}+{b}_{n}\right)}^{2}}}$

 for n=0,1,2,3,... (See arithmetic-geometric mean iteration.)
 • The GaussAGM of two positive real numbers a and b lies between their arithmetic mean

$\frac{a}{2}+\frac{b}{2}$

 • and their geometric mean

$\sqrt{ab}$

 • Each step of the iteration used to compute the GaussAGM computes an arithmetic mean and a geometric mean which explains the name GaussAGM. Lagrange discovered the arithmetic geometric mean before 1785.  Gauss rediscovered it in the 1790s and Gauss and Legendre developed the most complete theory of its use. GaussAGM can also be defined in terms of elliptic integrals
 > FunctionAdvisor( definition, GaussAGM(a,b))[1];
 ${\mathrm{GaussAGM}}{}\left({a}{,}{b}\right){=}\frac{{\mathrm{\pi }}{}\left({a}{+}{b}\right)}{{4}{}{\mathrm{EllipticK}}{}\left(\sqrt{\frac{{\left({a}{-}{b}\right)}^{{2}}}{{\left({a}{+}{b}\right)}^{{2}}}}\right)}$ (1)
 Note: The relation with elliptic integrals is discussed in the book Pi and the Arithmetic Geometric Mean by J. M. Borwein and P. B. Borwein (John Wiley & Sons).

Examples

 > $\mathrm{GaussAGM}\left(2.0,3.0\right)$
 ${2.474680436}$ (2)

 See Also