0
$\begingroup$

Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here.

Then $$ f(x) = \sum_n a_n x^n ; $$ assume also that $a_n\not= 0$ for all $n\ge 0$.

Let $$ A = \inf \sum_n a_n^2 , $$ with the infimum taken over all such $f$.

Is there a unique $f$ for which $A$ is assumed?

Improve this question
$\endgroup$
9
  • 2
    $\begingroup$ Are you sure there exists any such $f$? $\endgroup$
    – Daniel Weber
    Commented Jan 27 at 13:08
  • $\begingroup$ It is necessary to state what assumptions are made. This question just states a functional equation without any indication of the fact that this is an assumption. $\endgroup$
    – Daniel Asimov
    Commented Jan 27 at 15:44
  • 2
    $\begingroup$ @CommandMaster: This indeed seems quite non-trivial, but see this discussion here: en.wikipedia.org/wiki/Tetration#Complex_heights $\endgroup$
    – Christian Remling
    Commented Jan 27 at 16:22
  • $\begingroup$ @CommandMaster yes ! Many analytic solutions to tetration exist. Kneser for instance. $\endgroup$
    – mick
    Commented Jan 28 at 0:44
  • 3
    $\begingroup$ I'm talking about: Are you claiming that there exists an analytic function f(z) satisfying your conditions on the open disk |z| < 2 ? Are you claiming that your conditions define f(z) uniquely? Or are you asking these things in a question? It is not clear what role your conditions play. $\endgroup$
    – Daniel Asimov
    Commented Jan 28 at 1:39

0

Reset to default

You must log in to answer this question.