08/15/2023, 05:50 PM
(This post was last modified: 08/15/2023, 08:47 PM by 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.
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.
Regards
Shanghai46
Shanghai46
