How to find $f(x,y) = f(f_1(x,y),f_2(x,y)) $ ?
Notice that Gauss found the case $f_1 =( x + y)/2 , f_2 =( xy )^ {1/2}$. The socalled AGM. He even had an integral representation for it.
Even fractional iterations are possible
See
https://en.m.wikipedia.org/wiki/Arithmetic–geometric_mean
And
http://planetmath.org/convergenceofarithmeticgeometricmean/
Where the ratio is useful.
I have no idea how he came Up with the integral.
How to generalise the integral ?
How to handle the generalizations ?
Say i accept integrals and Abel funtions and inverse Abel functions, can i express it then ?
To clarify :
See : https://en.m.wikipedia.org/wiki/Abel_equation
Where the Abel function is a solution to the Abel equation.
Also
The inverse Abel function = superfunction