Recently I found out about Carlson elliptic integrals, which have great symmetry properties and allow to compute every kind of elliptic integrals and other functions.
The question is about the method which allows to compute one of the integrals:
$$R_F(x,y,z)=\frac{1}{2} \int_0^{\infty} \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}$$
To compute the integral we need the following 'mean values':
$$A_0=\frac{x+y+z}{3}$$
$$\lambda_0=\sqrt{xy}+\sqrt{yz}+\sqrt{zx}$$
(Actually, $A_0$ is just the arithmetic mean, and it's only used to obtain a better approximation for the integral).
Then we take $x_0=x$, $y_0=y$, $z_0=z$ and do the following iterations:
$$A_n=\frac{A_{n-1}+\lambda_{n-1}}{4}$$
$$x_n=\frac{x_{n-1}+\lambda_{n-1}}{4},~~~~~~y_n=\frac{y_{n-1}+\lambda_{n-1}}{4},~~~~~~z_n=\frac{z_{n-1}+\lambda_{n-1}}{4}$$
Finally we get:
$$\lim_{n \to \infty} A_n=\lim_{n \to \infty} x_n=\lim_{n \to \infty} y_n=\lim_{n \to \infty} z_n=\mu$$
$$R_F=\frac{1}{\sqrt{\mu}}$$
The question is:
Is $\mu$ some kind of a mean of the three numbers $(x,y,z)$?
How is it related to the arithmetic-geometric mean (AGM) of two numbers (which is used to compute 'classical' elliptic integrals)?