Tetration

There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.

Owing in part to some shared terminology and similar notational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:

There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other hyperoperations, while some are limited to tetration and have no immediate extension.

One notation above uses iterated exponential notation; this is defined in general as follows:

There are not as many notations for iterated exponentials, but here are a few:

Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate.

When evaluating tetration expressed as an "exponentiation tower", the serial exponentiation is done at the deepest level first (in the notation, at the apex). For example:

Smaller negative values cannot be well defined in this way. Substituting −2 for k in the same equation gives

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. There have, however, been multiple approaches towards the issue, and different approaches are outlined below.

To find a more natural extension, one or more extra requirements are usually required. This is usually some collection of the following:

A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:

Beyond linear approximations, a quadratic approximation (to the differentiability requirement) is given by:

The requirement of the tetration being holomorphic is important for its uniqueness. Many functions S can be constructed as

The extension of tetration into the complex plane is thus essential for the uniqueness; the real-analytic tetration is not unique.

One of the simpler and faster formulas for a third-degree super-root is the recursive formula, if: "x ^ x ^ x = a", and next x (n + 1) = exp (W (W (x (n) * ln (a)))), for example x (0) = 1.