meromorphic idea - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- 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