[FOM] Analyticity of half-exponentials

Timothy Y. Chow tchow at alum.mit.edu
Thu Apr 19 15:02:27 EDT 2007

Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:
>  "For the equation
> (*)              f^2(x) = e^x,
> a real analytic solution has been found by H. Kneser.
> This solution, however, is not single-valued (Baker)
> and, as pointed out by G. Szekeres, there is no
> uniqueness attached to the solution. It seems reasonable
> to admit f(x)=F^(1/2)(x), where F^u is the regular
> iteration group of  g(x)=e^x, as the "best" solution of
> the equation (*) (best behaved at infinity). However,
> we do not know whether this solution is analytic for x>0.

After taking a look at Baker's paper, I can parse the above better.  
The first sentence is the only one directly relevant to Joe Shipman's 
question, and answers it positively.  The rest of what Kuczma says here is 
concerned with *complex* solutions of f^2(x) = e^x.  In particular the 
terms "single-valued" and "analytic" (in the last sentence) refer to 
attempts to analytically continue f to the complex plane.


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