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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

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    IIRC there was an extensive discussion of "compound means" in the Borweins' *Pi and the AGM*. Did you check the book and see if your compound mean was treated there?2012-05-11
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    @J.M: No I did not. Is there any link to read it? Thanks for your references and advices.2012-05-11
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    I don't know of any link for reading it freely. Try checking the nearest university library.2012-05-11
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    @J.M:I found in amazon the book. It looks great. I am checking the limited read pages now. http://www.amazon.com/AGM-Computational-Complexity-Mathematical-Monographs/dp/047131515X2012-05-11
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    This question is somewhat similar in the sense that it also creates iterations of arithmetic mean and square-root mean (a.k.a. quadratic mean), although in a somewhat different manner: [Limit of the sequence $a_{n+1}=\frac{1}{2} (a_n+\sqrt{\frac{a_n^2+b_n^2}{2}})$ - can't recognize the pattern](http://math.stackexchange.com/questions/1826992/limit-of-the-sequence-a-n1-frac12-a-n-sqrt-fraca-n2b-n22).2016-08-25

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