Mizugadro, pentation, Book

[Kouznetsov]

Im fascinated by the (subset) Taylor(x,ln(x)) (or transseries if you want) used for the parabolic fixpoint.

Its ring structure makes solving the zoom equation ( F(f(x)) = f(ax) ) natural.

( polynomials of Taylor(x,ln(x)) are of same type and also Taylor(ax,ln(ax)) is of same type ! )

This needs more attention imho. ..

Thank you for your interest, tommy1729.
Have you write some similar expansions? Do they work?

Are all parabolic fixpoint expansions (for iterating analytic functions) like this ??

I think so. You may compare it to expansion of superfunction of transferfunction zex(z)=z exp(z)
It is described in chapter 12 of book "Суперфункции".
The Mathematica code to caluate the expansion is suggested in section 12.2.

.. what about terms ln(x)^7 x^3 then ?

You may extract the coefficient from the C++ implementation at
http://mizugadro.mydns.jp/t/index.php/E1etf.cin

Best regard, Dmitrii