(12/23/2014, 11:31 PM)tommy1729 Wrote: ....function for which every real iterate " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ).

...

=> with

=>

=>

Solve .. =>

Thus :

....

Hey Tommy,

Not sure I understood all of that ... But it inspired me to consider the following sequence of functions

...

Does g(x) converge, and is it a solution of interest to Mick? If g(x) converges, and it is analytic, then it has a Taylor/Laurent series....

Update:, by brute force, using a lot of computer cycles to estimate the limit, and then turn the coefficents it back into a fraction with power's of 2's... I get the following Laurent series, as the function that Mick might be looking for.

It would probably be normally expressed as

update2:
This would be compactly expressed via the Abel function as:

And then we get:

Finally, Mick's desired function in closed form would be as follows. With a little algebra, we generate all of the fractional iterates of g(z) as well. Then, using Mick's notation we have the desired g(z,t) function, which has all fractional iterates defined as:

for t=1, this is the same as the Laurent series above