Constructing Tetration as a flip on its head JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 07/14/2022, 12:30 AM (This post was last modified: 07/14/2022, 12:49 AM by JmsNxn.) I'm curious about any literature regarding a tetration function: $\text{tet}_b(z)$, such that it is your typical tetration function for $b \in \mathfrak{S}$, the Shell-Thron region, and in the complex plane uses traditional Schroder mechanisms. But then there exists a branch discontinuity at $b \in (\eta,\infty)$. I'm currently encountering this, and I'm wondering if anyone has any literature on this. Paulsen has little description of this, he just points out his solution has a branching problem at $\eta$, but he describes the branch cut along $(1,\eta)$, rather than the other way around. Theoretically this is entirely possible, because the fixed points move continuously from the Shell-thron region $|\log(y)| \le 1$ to $|\log(y)| \ge 1$, excluding the branching point at $e$, where as soon as you grow from here the fixed point pairs force a discontinuous choice. Largely because we must have real valued solutions, and any perturbation which is holomorphic must continue to be real valued. To be clear, I'm looking for a tetration solution: $\text{tet}_b(z)$ such that this function is holomorphic for $b \in \mathbb{C}/B$ where $B = (-\infty,e^{-e}] \cup [\eta,\infty)$. I've developed enough numerical evidence to suggest this should be a function, but I'm not entirely sure as of yet if it's a viable construction. Any help or comments is greatly appreciated. « Next Oldest | Next Newest »

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