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+--- Thread: closed form for regular superfunction expressed as a periodic function (/showthread.php?tid=502)
RE: closed form for regular superfunction expressed as a periodic function - bo198214 - 09/09/2010
(09/08/2010, 06:55 PM)tommy1729 Wrote: thus it is complex continuous ??
well its analytic. I guess can be continued to the whole complex plane, except one ray starting from the fixed point.
Quote:ive been thinking that entire functions with parabolic fixpoints with n somewhere analytic solutions for their half-iterate are half-iterates of another related function that has exactly n non-parabolic fixpoints an no other fixpoints or exactly n parabolic fixpoints with analytic solutions at their fixpoints and no other fixpoints.
Well not exactly like that rather it is the limit of a functions with n fixed points. In the case of e^x-1 or e^(x/e) this number is 2, and it is the limit of the regular iteration at the both real fixed points for b<e^(1/e). Generally it we consider the powerseries at the fixed point (0) and it is of the form \( x + a_m x^m + a_{m+1} x^{m+1}+\dots \) then the number of solutions is 2*(m-1). The domains of these solutions are arranged around the fixed point and give the so called Leau-Fatou-flower.
Quote:as the relationship between eta^x and e^x - 1.
dunno what you mean. Both have 1 parabolic real fixed point.
RE: closed form for regular superfunction expressed as a periodic function - tommy1729 - 09/09/2010
(09/09/2010, 10:12 AM)bo198214 Wrote: [quote='tommy1729' pid='5223' dateline='1283968558']
thus it is complex continuous ??
well its analytic. I guess can be continued to the whole complex plane, except one ray starting from the fixed point.
[quote]
ray ? you mean a line ? not ? a straigth line ? in what direction ? and is that a branch cut connecting the 2 solutions then ?