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What Lies Beyond Exponentiation?
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periods of tetration around real axis

Overview

Tetration by Escape Tetration by Period
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. A one page overview of my research can be found at Math Equation.

Paper in peer review

A New Kind of Science: Open Problems & Projects - Page 33
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The symbolic forms of the Ackermann function with a fixed first argument seem to have obvious interpretations for arbitrary real or complex values of the second argument. But is there a general way to extend these kinds of recursive definitions to continuous cases? Given a way to do this, how does it apply to recursive definitions like those on page 130? What happens to all the irregularities when one is between integer values? Or is it only possible to find simple continuous generalizations to functions that show fundamentally simple behavior? Can this be used as a characterization of when the behavior is simple? | | NKS page: 906 | Fields: functional analysis; recursive function theory

My paper The Existence and Uniqueness of the Taylor Series of Iterated Functions has been rejected by the Annals of Mathematics. While I disagree with a number of issues in the referee's report, I think it may also contain important perspectives on iterated functions that folks in the tetration research community have been unaware of. Referee’s report on The Existence and Uniqueness of the Taylor Series of Iterated Functions.

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