09/09/2010, 10:12 AM

(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 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.