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fractional iteration by schröder and by binomial-formula - 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: fractional iteration by schröder and by binomial-formula (/showthread.php?tid=719) |
fractional iteration by schröder and by binomial-formula - Gottfried - 11/23/2011 Just a small result from some playing around in a break; I'll look at the confirmation for a wider range of parameters later. This days I'm re-considering the differences of the fractional iterates of If we compare the according fractional iterates which we get using the Newton-formula (the binomial composition of integer-height-iterates only) which does not use a fixpoint, then it seems, that the results approximate that of the powers series around zero. I think this means for more general bases (other than 2): around the attracting fixpoint, which is zero only if log(base)<1 but is the negative real fixpoint if log(base)>1. The point x0, at which the most symmetric sinusoidal curve occurs seems to be x0~0.382160520000... (not rational!) which has a special property for base b=2 which I'll discuss in a later post. Gottfried |