(05/11/2009, 12:55 PM)sheldonison Wrote: Are \( F_{2,3} \) and \( F_{4,3} \) the same two functions in the Bummer post?
yes. They are also related to the earlier post which considers the two regular half iterates of \( \exp_{\sqrt{2}} \) on the interval (2,4), these are:
\( {\exp_{\sqrt{2}}}^{[1/2]}(z)=F_{2,3}(1/2+F_{2,3}^{-1}(z)) \)
and
\( {\exp_{\sqrt{2}}}^{[1/2]}(z)=F_{4,3}(1/2+F_{4,3}^{-1}(z)) \).
Quote:Could you comment on how the behavior of the two functions differ in the complex plane?
... Do they have the same periodicity?
... Does only one have singularities?
\( F_{4,3} \) is entire, has period \( 2\pi i/\ln(2\ln(2)) \).
\( F_{2,3} \) is not entire, has period \( 2\pi i/\ln(\ln(2)) \).
Dmitrii can perhaps tell more about the singularities.
Quote: Do both functions have the same values at z=+/-i*infinity?They have no limit along the imaginary axis because they are imaginary periodic.
Quote:Given that \( F_{2,3}(z)= F_{4,3}(z) \) at all integer values of z, then can these two functions be expressed in terms of each other, where \( F_{2,3}= F_{4,3}(x+\theta(x)) \)? Is the \( \theta(x) \) function analytic?
Yes, \( \theta(z)=F_{4,3}^{-1} ( F_{2,3}(z)) -z \) is analytic, though may somewhere have non-real singularities.