Uniqueness of half-iterate of exp(x) ?

[jaydfox]

Or maybe it's just plain wrong. A test using Matlab suggests that the f(x) I provided is not the solution. That's so weird; It seemed airtight to me, but there must be some pathology in my derivation. Hopefully someone with a better understanding of the W function can spot it for me.

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

Let's say f is of the following form,
\( f(x)=e^{bx} \)

Then the original definition becomes
\( f(f(x))=e^{be^{bx}}=e^x \)

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

Then your second equation becomes:
\( f(f(x))=e^{b(x)e^{b(x)x}}=e^x \)

But this is incorrect. The correct expansion is thus:
\( f(f(x))=e^{b(e^{b(x)x})e^{b(x)x}}=e^x \)

From there, the rest of the derivation is wrong.

BTW, it looked suspicious to me at first, because I knew b wasn't constant, but otherwise the derivation looked good, so I couldn't spot the error right away.