f( f(x) ) = exp(x) solved ! ! !

[andydude]

the equation \( f(f(x))=\exp(x) \) was not yet really solved, though we collected several approaches on that question on the forum.

Agreed. I should have said "has several approaches" instead of "solved".

What can be solved by regular iteration is \( f(f(x))=\exp(x)-1 \), though there are convergence issues, I think this is what you refer to.

Well, because \( F(x) = w^x - 1 \) and \( G(x) = w^{x/w} = e^x \) are topologically conjugate (where \( w = -W(-1) \)), I kind of skipped a step and thought of F (which has a fixed point of 0) and G (which has a fixed point of w) at the same time.

Also, I do understand the general idea of tommy's method, to make a function with a fixed point at 0, as a limit of a function with a fixed point not at zero. It makes sense. I just need to sit down at look over the math for a week or so, to convince myself that it all works, and that it gives new insight to the problem.

Andrew Robbins