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

[tommy1729]

Hey andydude,

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

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.

The observation of tommy was that there are methods (regular iteration!) to solve \( f(f(x))=F(x) \) for \( F(0)=0 \). So he just modified exp by subtracting the hump \( \exp(-nx^2) \) of height 1 which makes it 0 at 0. If you increase the \( n \) the hump gets sharper and sharper. For sufficient large \( n \) nearly everywhere except in the direct vicinity of 0 the modified function looks like \( \exp \).

Now he can take the regular half iterate of the modified function and hope that the modified functions will converge for increasing \( n \) and he would call the result then the half iterate of \( \exp \).

thank you bo for the clarification.

note that this can also be used to compute

f(f(f(x))) = exp(x).

and basicly f ^ (m/n) = exp(x)

which has a huge impact on tetration.


regards

tommy1729