Tetration
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About the Identities


         It is my hope that this list is as comprehensive as possible. To that end, I have included some things that have little to do with tetration or hyper-operations, except that they are nice relationships. As a general note, some browsers have trouble with multiple subscripts or superscripts, so where there might be a problem I have either made the formula into an image, or used either "_" or "^" respectively.

Axioms


         •           the defining property of tetration
         •           the defining property of super-logarithms

Identities

Common Definitions

         • nxa =          for iterated exponentiation
         • nx = for integer tetration

Simple Definitions

Inverse Function Formulas

Integer Values of Tetration

Integer Values of Iterated Exponentiation

The Infinite Iterated Exponential and Company

This table shows an interesting relationship: that these functions are so similar, they can all be expressed in terms of each other.

         • x = −W(−log(z)) / log(z)         = 1 / srt2(1 / x)          the infinite iterated exponential
         • (ex)x = W(x) = x / srt2(ex)          the product-logarithm
         • 1 / (1 / x)         = log(z) / W(log(z)) = srt2(x)          the square super-root


General Piecewise-Definitions

For tetration:          where t(x, y) is the critical function for tetration.

For the super-logarithm:          where s(x, z) is the critical function for the super-logarithm.

A Simple Extension to Real Numbers

The simplest extension to real numbers is the bare minimum required to obey the known integer values for tetration and the super-logarithm: which satisfy the constraints:          and

Complex Conjugate Properties

Using my extension, these have been verified numerically for inputs with small imaginary parts between −1/2 and 1/2.

Integration Formulas

Tetration Notation

Iterated Exponentiation Notation

Galidakis uses the notation y(x, a) for iterated exponentiation, which is beneficial when a is complicated. In ASCII, the same can be expressed using the notation "x^^y@a", which can be found in several places around the internet. Interestingly, some have noticed that the notation "x^^y.a" is equivalent to the first-degree approximation of tetration, where y is the integer part and a is the fractional part of the value being tetrated.

Miscellaneous

Other Things Known



Visited: 3140 times, last updated: 2006-02-15, by: Andrew Robbins, contact: and_j_rob(at)yahoo(dot)com