Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible?
if not possible, what is the closed form of it as known functions ?
$AGM=M(x,y)=\frac{\pi}{4}\frac{x+y}{K(\frac{x-y}{x+y})}$
where $K(m)$ is the complete elliptic integral of the first kind:
$K(m)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{1-m^2\sin^2(x)}}$
Iterative Root-Mean Square-Arithmetic Mean calculation:
$r_1=\sqrt{\frac{r_0^2+a_0^2}{2}}$
$a_1=\frac{r_0+a_0}{2}$
$r_{n+1}=\sqrt{\frac{r_n^2+a_n^2}{2}}$
$a_{n+1}=\frac{r_n+a_n}{2}$
Root Mean Square-Arithmetic Mean of $(r_0,a_0)=RMSAM(r_0,a_0)=\lim\limits_{n\to \infty} r_{n}=\lim\limits_{n\to \infty} a_{n}$
Thanks a lot for answers