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

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

and their geometric mean

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];

$\mathrm{GaussAGM}\left(a\,b\right)=\frac{\mathrm{\pi }\left(a+b\right)}{4\mathrm{EllipticK}\left(\sqrt{\frac{{\left(a-b\right)}^2}{{\left(a+b\right)}^2}}\right)}$

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).

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

$2.474680436$