Tetration

It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.

The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.

Note that nested exponents are conventionally interpreted from the top down: 357 means 3(57) and not (35)7.

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:

In the first two expressions a is the base, and the number of times a appears is the height (add one for x). In the third expression, n is the height, but each of the bases is different.

Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials.

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:

This order is important because exponentiation is not associative, and evaluating the expression in the opposite order will lead to a different answer:

Since complex numbers can be raised to powers, tetration can be applied to bases of the form z = a + bi (where a and b are real). For example, in nz with z = i, tetration is achieved by using the principal branch of the natural logarithm; using Euler's formula we get the relation:

This suggests a recursive definition for n+1i = a′ + b′i given any ni = a + bi:

Solving the inverse relation, as in the previous section, yields the expected 0i = 1 and −1i = 0, with negative values of n giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit 0.4383 + 0.3606i, which could be interpreted as the value where n is infinite.

Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.[citation needed]

As the limit y = x (if existent on the positive real line, i.e. for eexe1/e) must satisfy xy = y we see that xy = x is (the lower branch of) the inverse function of yx = y1/y.

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

which is not well defined. They can, however, sometimes be considered sets.[9]

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:

The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights; one is based on the regularity requirement, and one is based on the differentiability requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.

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

The proof is that the second through fourth conditions trivially imply that f is a linear function on [−1, 0].

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

Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree n also exist, although they are much more unwieldy.[1][13]

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

where α and β are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of Im z.

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.

Other than the problems with the extensions of tetration, there are several open questions concerning tetration, particularly when concerning the relations between number systems such as integers and irrational numbers: