# Main Page

## Overview

My name is Daniel Geisler, welcome to my web site dedicated to promoting research into the question of what lies beyond exponentiation - tetration, the hyperoperators and the Ackermann function.

The original tetration.org site is now at tetration.org/original.

Extending tetration and the Ackermann function to the complex numbers fractals

### Tetration

Tetration is iterated exponentiation, written as . A major math problem that many amateur mathematicians find their way to is the question of how to extend tetration from the whole numbers , to reals and complex values . The expression is easy to compute, but expressions like and becomes very difficult, but not impossible.

### Iterated Functions

### Ackermann Function

Exponentiation , tetration , pentation and on with the higher hyperoperators.

## Fractional iteration with code

flows

, where is a fixed point

the Lyapunov multiplier, denoted , with .

### Mathematica code

Flow[f_, t_, x_, L_, order_ : 3] := Module[{}, H[0] = L; H[1] = f'[L]^t ; Do[ H[max] = First[r[t] /. RSolve[{r[0] == 0, r[t] == Sum[ Derivative[k][f][L] BellY[max, k, Table[H[j] /. t -> t - 1, {j, max}]], {k, 2, max}] + f'[L] r[t - 1]}, r[t], t]], {max, 2, order}]; Sum[1/k! H[k] (x - L)^k, {k, 0, order}] ]; L=ProductLog[-Log[x]]/Log[x];