Constructive real tetration Daniel Fellow Posts: 240 Threads: 78 Joined: Aug 2007 08/31/2022, 12:00 PM Ouch, I've been bitten by the real tetration bug! I am writing Mathematica software that includes my own approach to complex tetration, but I want to support other approaches to tetration, particularly real tetration. What constructive methods are there out there for extending tetration? I do plan on adding support for Gottfried's work. Daniel JmsNxn Ultimate Fellow Posts: 1,064 Threads: 121 Joined: Dec 2010 09/01/2022, 05:41 AM (This post was last modified: 09/01/2022, 05:41 AM by JmsNxn.) Between $$1 < \alpha < \eta$$, we can construct tetration entirely from a real valued iterated integral. The algorithm is written firstly with the identification: \begin{align} \alpha \uparrow^0 x &= \alpha \cdot x\\ \alpha \uparrow^n \alpha \uparrow^{n+1} x &= \alpha \uparrow^{n+1}\,x+1\\ \alpha \uparrow^n 1 &= \alpha\\ \alpha \uparrow^{n+1} k &= \alpha \uparrow^n \alpha \uparrow^n \cdots k\,\text{times}\cdots \uparrow^n \alpha \end{align} Then there's a Picard type integral sequence to generating this construction: \begin{align} \vartheta(w) &= \sum_{k=0}^\infty \left(\alpha \uparrow^{n} k+1 \right)\frac{w^n}{n!}\\ \Gamma(1-x) \left(\alpha \uparrow^n x\right) &= \sum_{j=0}^\infty \left(\alpha \uparrow^n k+1\right) \frac{(-1)^k}{k!(k+1-x)} + \int_1^\infty \vartheta(-w)w^{-x}\,dw \\ \end{align} Recalling that $$\alpha \uparrow^{n} k+1$$ is just $$\alpha \uparrow^{n-1} \cdots \alpha \uparrow^{n-1} \alpha$$ with $$k+1$$ alphas. Where this defines a recursive protocol. I have it written here on ArXiv https://arxiv.org/abs/2106.03935; which details in much more rigor, why this holds. But this still just produces the standard regular iteration. So it's not a new "real tetration"--it's just a different approach. (Something I think is going to become much more relevant with all the stuff we're working on now with Borel sums!) The only other real tetration I have to offer, is the beta method. Which only produces smooth values for bases $$b > \eta$$, but can produce analytic values within the Shell thron region. I suggest, my release of the code--beta.gp--if you want to try and implement that in mathematica. It would be hell though. Because I have 48 gbs of ram on my laptop, and I program in GP with that knowledge. Which means a lot of my code is designed for large and robust amounts of value. Though it does work for simple stuff, I imagine it would be tenuous on Wolfram Alpha.... Even in a C shell my code is slow, lol. But if you wait for an hour after writing Sexp(1+z) you'll get the taylor series about $$z=0$$ of this function to ridiculous accuracy   And after shamelessly supporting my own work--you should look at Kouznetsov's C++ program. I can locate it down if you want, but I sent you his book, and I believe the link to it is in the book. « Next Oldest | Next Newest »

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