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

Joe Shipman joeshipman at aol.com
Fri Aug 28 18:18:30 EDT 2015

```I'm glad to hear that this is an active research topic. I'll follow up these links, but do you know the answers to any of the following?

Is there f with f(f(x))=e^x which is both real analytic and monotonic for x>0?

Is Kneser's construction generalizable to find monotonic real analytic g(g(x))=f(x) and so on?

Is it generalizable to find monotonic real analytic f(f(x))=2^x?

What kind of conditions on such f imply uniqueness?

I don't care at all if the function is not entire but if it is not monotonic I can't regard it as a proper extension of the "tower of exponents" function to non-integral arguments.

-- JS

Sent from my iPhone

> On Aug 28, 2015, at 11:50 AM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
>
>> 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
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```