User:Daniel Geisler

I am a dot com refugee with a broad interest in science and mathematics who has returned to college to pursue a degree in mathematics. I have a personal web site and a site devoted to tetration and the Ackermann function that has several dozen pages of content. The site would be much bigger if I had a most effective and flexibility way of publishing, so I’m currently running several Wiki pilot projects.

I have added a couple of tetration fractals to the tetration page, but I have other fractals I could donate if folks were interested. I am writing an article on tetration for publication. I’ve added some historical background with supporting references to the tetration and Ackermann function pages. I'm involved with the informal community of folks who respond to questions about tetration on the Internet.

Tetration links

Mathematics brain teasers Ioannis Galidakis does extensive research on tetration. Galidakis’s web site contains the definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.
Web pages for infinitely iterated exponentials Dave L. Renfro has compiled entries from questions about tetration on sci.math.
Extension of the hyper4 function to reals Robert Munafo discusses extending tetration to the real numbers.

Daniel Geisler 03:38, 21 Feb 2005 (UTC)

Thoughts on the "Extension to low values of the second operand"

Using the relation $n\uparrow\uparrow k = \log_n \left(n\uparrow\uparrow (k+1)\right)$ (which follows from the definition of tetration), one can derive (or define) values for $n\uparrow\uparrow k$ where $k \in {-1, 0, 1}$ .

$\begin{matrix} n\uparrow\uparrow 1 & = & \log_n \left(n\uparrow\uparrow 2\right) & = & \log_{n} \left(n^n\right) & = & n \log_{n} n & = & n \\ n\uparrow\uparrow 0 & = & \log_{n} \left(n\uparrow\uparrow 1\right) & = & \log_{n} n & & & = & 1 \\ n\uparrow\uparrow -1 & = & \log_{n} \left(n\uparrow\uparrow 0\right) & = & \log_{n} 1 & & & = & 0 \end{matrix}$

Tetration just like another mathematics needs to have an axiomatic basis. The trick is to properly enumerate the list of possible consistent systems which extend tetration beyond the positive integers. $n\uparrow\uparrow k,$ $k > 0$ is computed by iterated exponentiation is standard.

$n\uparrow\uparrow k,$ $k < 0$ can extend k into the negative integers by using iterated logarithms instead of iterated exponentiation, but this means that $n\uparrow\uparrow k$ becomes multivalued.

This confirms the intuitive definition of $n\uparrow\uparrow 1$ as simply being $n$ . However, no further values can be derived by further iteration in this fashion, as $log n 0$ is undefined.

Articles on arithmetic including treat $n\uparrow\uparrow 1$ as an axiom, not something capable of comfiming an intuition. Furthermore, the dynamics of the Riemann sphere have no problem with dealing with $log n 0$ or the further logarithmic iterations from zero.

Similarly, since $log 11$ is also undefined ( $log 1 1 = ln1 / ln1 = 0 / 0$ ), the derivation above does not hold when $n = 1$ . Therefore, $1\uparrow\uparrow{-1}$ must remain an undefined quantity as well. (The figure $1\uparrow\uparrow{0}$ can safely be defined as 1, however.)

Again, $00$ is an undefined quantity, so values for $0\uparrow\uparrow{k}$ cannot be defined directly. However, $\lim_{n\rightarrow0} n\uparrow\uparrow{k}$ is well defined, and exists:

$\lim_{n\rightarrow0} n\uparrow\uparrow k = \begin{cases} 1, & k \mbox{ even} \\ 0, & k \mbox{ odd} \end{cases}$

This limit holds for negative $n$ , as well. $0\uparrow\uparrow{k}$ could be defined in terms of this limit, but $0\uparrow\uparrow2 = 0$ would conflict with the standard undefinedness of $00$ .