[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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