Extended Schroeder equation

[Shanghai46]

So, I rencently realized the link between my tetration method, and Schroeder's equation. The thing is, my formula is somewhat different from Schroeder's.

Basically, the normal version is like this : \(Ψ(f(x))=sΨ(x)\).

And in that case, for any real or complex number k, the kth iteration of f(x) is \(Ψ^{ -1}(Ψ(x)s^k)\).
If \(a\) is a hyperbolic attractive fixed point equal to 0, then \(s=f'(a)\).
My question is : Is my "extended version" of that equation correct?

Which is if \(a\) is a hyperbolic attractive fixed point, with \(f'(a)=s\), then : 

\(Ψ(f(x))-a=s(Ψ(x)-a)\)
Then for any real or complex k, \(f^k(x)=Ψ^{-1}((Ψ(x)-a)s^k+a)\)
AND \(Ψ(x)=\lim_{n \to + \infty}(f^n(x))\)

I have every reason to believe this to be true, but I would prefer other people to check that.

[leon]

Yes your formula works and is good to know but could look a little syntax dense

a Schroeder equation is like an extended forray (like going to an exotic club or Mexican restaurant) into integrating things like Riemann sums and outderiving a variable. So because of things like iterates and dual triple and quadruple (like tetration) iterates, or gaussian strings and higher order derivatives, the Schroeder equation should be easy enough to tackle and master

it saves the trouble of isolating a variable like using a different Riemann sum by doing it with a factor

if you're having trouble remembering Schroeder just remember to make the bigger part of the Riemann sum bigger than a (nother variable

your version you could remember all the topological fields that use the symbols