Overview

Tetration by Escape	Tetration by Period

My name is Daniel Geisler , welcome to my web site dedicated to promoting research into the question of what lies beyond exponentiation. Tetration is defined as iterated exponentiation but while exponentiation is essential to a large body of mathematics, little is known about tetration due to its chaotic properties. The standard notation for tetration is \(^{1}a=a,   ^{2}a=a^a,  ^{3}a=a^{a^a},\) and so on. Mathematicians have been researching tetration since at least the time of Euler but it is only at the end of the twentieth century that the combination of advances in dynamical systems and access to powerful computers is making real progress possible.

The big question in tetration research is how can tetration be extended to complex numbers. How do you compute numbers like \(^{.5}2\), and \(^{\pi i}e\) ? This web site will show how to compute these and other problems. See the Tetration page for a one page overview of extending tetration to the complex numbers.

Changes in Tetration.org

Tetration.org was originally designed and written in a six week period six years ago. In that time a revolution in tetration research has occurred. In that time a revolution in tetration research has occurred. Ioannis Galidakis has published a number of papers on tetration and a growing community of talented mathematicians collaborates on tetration research through sci.math.research and the Tetration Forum. Still, the central goal of publishing a universally accepted and understood extension of tetration to the real and complex numbers hasn’t been achieved yet. A number of algorithms have been proposed for extending tetration, but they have have not been proven to converge, be unique and be consistent. Hopefully I can present proofs of these properties in the near future.

The most obvious change to Tetration.org is that it is now using the MathJax JavaScript package to display \(\LaTeX\).

Convergence of Iterated Functions

Theorem. Let \(f(z)\) be a uniformly convergent function. Then \(f^n(z), n \in \mathbb{N}\) is also uniformly convergent.

Let \(f(z)\) and \(g(z)\) be a uniformly convergent functions. Then \(f(g(z))\) is also uniformly convergent. By induction \(f(f(z))=f^2(z), f(f^2(z))=f^3(z), \ldots, f(f^{n-1}(z))=f^n(z)\), therefore \(f^n(z), n \in \mathbb{N}\) is also uniformly convergent.

Theorem. Let \(f(z)\) be a uniformly convergent function. Then \(f^m(z), m \in \mathbb{Q}\) is also uniformly convergent.

If \(m \in \mathbb{Q}\) then there exists a \(k \in \mathbb{Z}\) such that \(k m = n \in \mathbb{Z}\).