I no longer focus exclusively on tetration, I mostly work with smooth complex iterated functions \( f^n(z) \) to serve my attack on hyperoperations. Because I didn't have anyone to work with, I had to devise methods to independently check my work. My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
My most general derivation only assumes there is a non-super attracting fixed point. It handles the entire complex plane - the Shell-Thron boundary, it's interior and external. My Mathematica software validated the sanity check out to \( \mathcal{O}(z^{8}) \). The only function with only a fixed point at infinity is the successor function which is trivial. My numerical computations indicate that my method can correctly compute the position of neighboring fixed points. Can any other tetration method correctly place the positions of the fixed points.
I have a admission to make. I don't trust any fractional iteration techniques, including those published, but my own and Carleman matrices, which are experimentally consistent with my work. I strongly suspect I am wrong in this attitude and I am trying to expand my math background with the material on this site. Unfortunately I only made it through sophomore year of college, so I am almost completely self-taught. For example, the Riemann mapping theorem was a pleasant surprise to me.
I would feel comfortable with these other techniques if they could pass my sanity test.
My most general derivation only assumes there is a non-super attracting fixed point. It handles the entire complex plane - the Shell-Thron boundary, it's interior and external. My Mathematica software validated the sanity check out to \( \mathcal{O}(z^{8}) \). The only function with only a fixed point at infinity is the successor function which is trivial. My numerical computations indicate that my method can correctly compute the position of neighboring fixed points. Can any other tetration method correctly place the positions of the fixed points.
I have a admission to make. I don't trust any fractional iteration techniques, including those published, but my own and Carleman matrices, which are experimentally consistent with my work. I strongly suspect I am wrong in this attitude and I am trying to expand my math background with the material on this site. Unfortunately I only made it through sophomore year of college, so I am almost completely self-taught. For example, the Riemann mapping theorem was a pleasant surprise to me.
I would feel comfortable with these other techniques if they could pass my sanity test.
Daniel