# Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{}$$\!^{} of 2 numbers a and b is denoted \operatorname{AGM}(a,b) and defined as follows:$$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,\quad b_{n+1}=\sqrt{a_n b_n}.\text{Then}\quad\operatorname{AGM}(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n.$$The arithmetic-geometric mean can be expressed in a closed form using the complete elliptic integral of the first kind and elementary functions. Let us try to generalize the arithmetic-geometric mean to 3 numbers a,b and c. One way to define it would be just as \operatorname{AGM}\left(\frac{a+b+c}3,\sqrt{abc}\right). Apparently, this gives us nothing really new or interesting. Let us consider a different approach:$$\text{Let}\quad a_0=a,\quad b_0=b,\quad c_0=c,\quad a_{n+1}=\operatorname{AGM}(b_n,c_n),\quad b_{n+1}=\operatorname{AGM}(a_n,c_n),\quad c_{n+1}=\operatorname{AGM}(a_n,b_n).\text{Then}\quad\operatorname{AGM}(a,b,c)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n=\lim_{n\to\infty}c_n.$$This gives us a function different than one in the previous approach. For example, we can calculate that$$\operatorname{AGM}(1,2,3)\approx1.909157449373156462538798818255615478726285889167...$$(you can see more digits here) Have this function and its properties been already studied? What is known about it? Is it possible to express$\operatorname{AGM}(a,b,c)$(or, at least, some of its non-trivial special values) in a closed form using known special functions? • I assume this is related to nesting complete elliptic integrals but I couldn't find anything on the subject. Elliptic rational functions was the closest thing I could find but I think they are still too different. – Ali Caglayan May 22 '16 at 1:22 • There is a generalization of AGM (see the cubic version at mathoverflow.net/q/202008/15540) but this one for three numbers is new. +1 for the same. – Paramanand Singh May 22 '16 at 11:41 • Could this be the same limit? math.stackexchange.com/questions/442062/… – alphacapture May 24 '16 at 19:26 • @alphacapture, no, it's not the same. For the question you referenced the limit for$1,2,3$will be$1.9099262335408153237$– Yuriy S Aug 18 '16 at 18:16 • Have you tried$(1,1,\sqrt{2})$or$(1,\sqrt{2},\sqrt{2})$since the only closed form for a classical AGM appears for$1$and$\sqrt{2}\$? – Yuriy S Aug 23 '16 at 12:25