Uniqueness of half-iterate of exp(x) ?

[jaydfox]

I think the problem is that b is not a constant. It is a function of x. That's where the math goes wrong:

The first equation should be written:
\( f(x)=e^{b(x)x} \)

Hmm, now that I think about it, you really can treat b as a constant. From your derivation, one of the steps is:

\( W(x^2)=bx \)

Given a value of x, we can solve for b:

\( W(x^2)=bx\\
\frac{W(x^2)}{x}=b \)

For example, if we let x = 1, then we find b=0.567143290409784
Then we validate:

let x = 1
let b = 0.567143290409784
exp(b*exp(b*x)) = 2.718... = exp(1)


As another example, let x=3. Then we find that b=0.559672139928533

Putting it together, exp(b*exp(b*x))=20.0855369231877 = exp(3)

So it works. The problem is, the value of b will vary with each value of x. If we make b constant, then it will only work for a small set of values of x (possibly a single value of x). Otherwise, we have to make b a function of x, and then the derivation does not work, as I previously pointed out.