Just a few late thoughts

[Leo.W]

1.)
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Which is slightly different than your form. I'm very interested in this. Additionally it was shown that in general, these solutions are not holomorphic (barring Schroder solutions). So it's no help for base \(e\) for example (at least real valued ones).

Hi James
Thx for your reply
I don't quite get your ideas about how your transformations work, could you show me more details?
Does it work by expanding F(z) as a power series in s^z where s is the multiplier? Just my intuition lol

Well for the 2nd part, it's a more general phenomenon, could be in Set theory. pardon me because I didnt explain why I jumped to this
Here are details (sry but i literly have so lil time) (and maybe someone proposed similar things way before)
After I posted about the iteration research for \(z+\Gamma(z)\), I realized that the conjugator couldn't be any elemental functions but more transcendental ones, most likely to be a function that has asymptotic behaviors of inverse function of \(\sqrt{\frac{C}{z\sinh(z)}}\), so I consider the computation of the inverse of \(z*sin(z)\), but it turned out that I should investigate for an asymptotic again, for this function.
Looked into the asymptotic of inverse questions, then after deriving the inverse asymptotic(mostly I refer to the one at infinity) of the function \(z+\frac{1}{z}+\log(z)\), I noticed and found it pretty easy to prove that:
For \(f(z)=z+\log(z)+O(\log(z))\), we must have \(f^t(z)=z+t\log(z)+O(\log(z))\)
And can be extended further and further such as:
For \(f(z)=z+G(z)+O(G(z)), where G(z)=O(z)\), \(f^t(z)=z+tG(z)+O(G(z))\)
And other cases like \(f(z)=H(z)+G(z)+O(G(z))\) where \(G(z)=O(H(z))\)
Thus we have a method for more series' generalization,
for example, if we consider the function \(f(z)=z+\sqrt{z}\), how would you calculate \(f^t(z)\)?
We can write by asymptotic \(f^t(z)=z+t\sqrt{z}+\frac{t(t-1)}{4}+O(z^{-1/2})\) and as more terms as desired.
And we use asymptotic expansion to get any iteration by conjugated with f.
If we only took fixed point method we won't get any f^t.
This asymptotic expansion has a group structure under composition, not necessarily at inf, for example the set
\(\{f(z)|asymp[f](z)=\sum_{n\ge -k}{P_n(log(z),log^2(z),\cdots,log^r(z)}z^{-\frac{n}{T}}\}\) forms a group under composition, where T is a rational number and r any positive integer and P_n any multidimensional rational(polynomial) function.
And all such group guarantee an expansion of iterations of any element inside of it, with t.
The very well-known example is the set of polynomials with 0 as constant term, or
\(\{f(z)|asymp[f](z)=\sum_{n\ge1}{a_nz^n}\}\)
And their iterations' series are well known too
This is only the beginning but i don have so much time though
What becomes next is beyond the structure, we must conquer a method of computing asymp from asymp to be more clear about the whole struct, suck like asking for an asymp for a superfunction of these functions.
That is, \(F(z+1)=F(z)+G(F(z))+O(G(F(z)))\)
You may ask why we'd take it into account, for example
You can know at first glance about
\(f(z)=z+\sqrt{z}\to f^t(z)=z+t\sqrt(z)+O(t^2z^0)\)
But what about asymp in t (eg. t->inf)?
\(f(z)=z+\sqrt{z}\to f^t(z)=\frac{t^2}{4}+O(t\log(t)h(z))\)
This can be a hard problem bro