# Tetration:Complex

### From Tetration.net

## Contents |

## Complex Tetration

The focus here is to derive . In general where , therefore if then can be found by taking and setting giving .

### Hyperbolic Tetration (First three terms)

Consider how to define hyperbolic tetration for where the function to be continuously iterated is and .

**Note**: Sorry about the formatting, this site isn't properly parsing certain TeX commands.

### Parabolic Tetration (First three terms)

Using the equation for parabolic continuous iteration, where and then setting gives,

### Rationally Neutral Tetration

#### Period 2

Consider the bifurcation point between period one and period two at . The exponential map has a derivative of - 1 at its fixed point. Therefore its second iterate has a derivative of 1 at its fixed point and can be solved as a case of parabolic continuous iteration.

#### Period n

In general if an exponential map has a first derivative that is a n^{th} root of unity at its fixed point, then the **n**^{th} iterate of the exponential function can be solved using parabolic continuous iteration.

### Superattracting Tetration