# 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 ${\displaystyle a=\,{}^{1}a,\;a^{a}=\,{}^{2}a,\;a^{a^{a}}=\,{}^{3}a,\cdots }$. A major math problem that many amateur mathematicians find their way to is the question of how to extend tetration from the whole numbers ${\displaystyle \{\,{}^{n}a\in \mathbb {Z} ^{+}\,|\,a,n\in \mathbb {Z} ^{+}\}}$, to reals ${\displaystyle \{\,{}^{n}a\in \mathbb {R} \,|\,a>e^{1/e},n\geq 0\in \mathbb {R} \}}$ and complex values ${\displaystyle \{\,{}^{n}a\in \mathbb {C} \,|\,a,n\in \mathbb {C} \}}$. The expression ${\displaystyle {}^{2}2=2^{2}=4}$ is easy to compute, but expressions like ${\displaystyle {}^{1/2}2}$ and ${\displaystyle {}^{i}e}$ becomes very difficult, but not impossible.

### Ackermann Function

Exponentiation ${\displaystyle a\uparrow b}$, tetration ${\displaystyle a\uparrow ^{2}b}$, pentation ${\displaystyle a\uparrow ^{3}b}$ and on with the higher hyperoperators.

## Fractional iteration with code

flows

${\displaystyle H(0,t)=L}$, where ${\displaystyle L}$ is a fixed point

${\displaystyle H(1,t)=f'(L)^{t}}$ the Lyapunov multiplier, denoted ${\displaystyle \lambda }$, with ${\displaystyle \lambda \neq 0}$.

${\displaystyle H(n,t)=\sum _{r=0}^{\infty }(\sum _{k=1}^{n}{\frac {f^{(k)}(L)}{k!}}B_{n,k}(H(1,t-1),\ldots ,H(n-k+1,t-1)))^{r}}$

${\displaystyle f^{t}(x)=\sum _{n=0}^{\infty }{\frac {1}{k!}}H(k,t)(x-L)^{k}}$

${\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}$

### 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];


Discrete and continuous tetration compared