> Is this kind of equation known? > ^[f(x)] means iteration f(x) times. > How do we 'deal' with such a thing?
Alain, expressions like f^{f(x)}(x) have a clear meaning when considered in the context of the Ackermann function.
Addition f(x) = a+x f^{n}(x) = a*n + x f^{f(x)}(x) = a*(a + x) + x
Multiplication f(x) = a*x f^{n}(x) = a^n * x f^{f(x)}(x) = a^{a*x} * x
See http://www.tetration.org/scimath/1 for an exported Mathematica notebook. While both addition and multiplication result in solutions for f(x), the plots of the solutions show they are not monotonic. More generally, f^{f(x)}(x) is meaningful when using continuously iterated functions, but then you loose the chance of arriving at a closed form solution. I would be very surprised if anyone could find a monotonic closed form solution. Daniel
Thank for your reply. I've got a different view and prefer standing heavily upon Abel counting functions. When phi(x) and f(x) are such as :phi(f(x))=phi(x)+1 phi(f^[r](x) = phi(x) + r ,r real ;with phi invertible f^[r](x) = phi^[-1](phi(x) +r) ;and with r=f(x) f^[f(x)](x) = phi^[-1](phi(x) +f(x)) . With this tool you can ALSO solve for instance: f((2x +1)^[f(x)])= 3*f(x) ... I am sure there is still much work for us!. COURAGE, Alain.
