[FOM] 605: Integer and Real Functions (Harvey Friedman)

[Timothy Y. Chow]

On Fri, 28 Aug 2015, Joe Shipman wrote:
> What do you mean "none that are holomorphic"?

Sorry, what I meant was, "none that are entire".  This was a MathOverflow 
question once.

http://mathoverflow.net/questions/12081/does-the-exponential-function-have-a-square-root

> Is there actually a theorem of the following form?
>
> "there is no real function f(x) such that
> (1) there is a nonempty interval (a,b) in the reals such that for all x 
> in (a,b), f(f(x))=e^x
> (2) for some open set W in the complex plane that includes the reals in 
> (a,b), there is a holomorphic extension of f defined on W."

No, in fact Kneser obtained a real-analytic solution to f(f(x)) = e^x.

Hellmuth Kneser, Reelle analytische Loesungen der Gleichung 
phi(phi(x))=e^x und verwandter Funktional-gleichungen, J. Reine Angew. 
Math. 187, (1949). 56-67.

Being real-analytic means that the Taylor series has a non-zero radius of 
convergence so it can be extended to a holomorphic neighborhood of (a,b).

Tim