[Update] Comparision of 5 methods of interpolation to continuous tetration

[tommy1729]

The only thing I can come up with is that :

1) we simply take the principal schroeder function near the fixpoint c.

2) in that domain we use analytic continuation ( monodromy ) so that by small radiuses recentered we can reach any value in the upper plane.

3) In that upper plane we find ( after many continuations ) the values equalling the reals between 0 and 1. ( and those are connected )
( for reasons yet to be explained !)

4) the functional equation still holds after the analytic continuation ( for reasons yet to be explained )

5) We do a Riemann mapping from those values ( reals between 0 and 1) onto the reals between 0 till 1. (we connect 1 to 1)

6) in 5) the functional equation still holds ( again for reasons yet to be explained since we are outside the radius of convergeance of c and did a mapping ! )

7) by swcharz reflection and analytic continuation we our desired real to real schroeder function.

by using log we get the desired abel function.

9) take the functional inverse of the abel function ( series reversion maybe ).

10) we have sexp analytic near the positive real line.

I probably got it wrong or didnt I ?

And if I got It right , there is still alot to be explained !!

regards

tommy1729