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+--- Thread: meromorphic idea (/showthread.php?tid=292)
meromorphic idea - tommy1729 - 05/13/2009
i was thinking about tetration and came up with important questions.
for the 3rd time !!
are there meromorphic functions f(x) so that they commute with exp(x) ?
thus :
exp(f(x)) = f(exp(x))
related 2nd question :
t ( t(x) ) = exp(x) ( any real analytic t(x) )
g(x) a meromorphic function
t(x) = inverse_g( 1-fixpoint-regular-half-iterate[ exp(g(x)) ] )
related 3rd question :
q(x) a meromorphic function
t ( t(x) ) = exp(x) ( any real analytic t(x) )
1-fixpoint-regular-half-iterate[ exp(q(x)) ] = t( q(x) )
thanks in advance
regards
tommy1729
RE: meromorphic idea - andydude - 05/13/2009
(05/13/2009, 04:52 PM)tommy1729 Wrote: are there meromorphic functions f(x) so that they commute with exp(x) ?
I'm not sure if this is important, but I think what is important is whether or not:
"For all f(x) that satisfy \( f(\exp(x)) = \exp(f(x)) \), there exists a unique real number t such that \( f(x) = \exp^t(x) \)."
I'm not convinced that this is always true for holomorphic/meromorphic functions. I'm sure its false for for arbitrary (or piecewise-defined) functions. I also think this would be useful in characterizing fractional iterates.
Andrew Robbins
RE: meromorphic idea - tommy1729 - 05/13/2009
andydude wrote :
(05/13/2009, 09:51 PM)andydude Wrote:(05/13/2009, 04:52 PM)tommy1729 Wrote: are there meromorphic functions f(x) so that they commute with exp(x) ?
I'm not sure if this is important, but I think what is important is whether or not:
"For all f(x) that satisfy \( f(\exp(x)) = \exp(f(x)) \), there exists a unique real number t such that \( f(x) = \exp^t(x) \)."
I'm not convinced that this is always true for holomorphic/meromorphic functions. I'm sure its false for for arbitrary (or piecewise-defined) functions. I also think this would be useful in characterizing fractional iterates.
Andrew Robbins
i kinda asked this question before - more or less -
see thread :
http://math.eretrandre.org/tetrationforum/showthread.php?tid=270
yes , this is useful in characterizing fractional iterates.
and i believe it is important for some half-iterate questions , though i cant prove it to be for tetration , but im working on it.
regards
tommy1729