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

Timothy Y. Chow tchow at alum.mit.edu
Fri Aug 28 11:50:03 EDT 2015

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.


> 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).


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