Tetration

In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation.

The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions.

) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of

) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving

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).

Although the base and the height can be extended beyond the non-negative integers to different domains, including

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: The following approximate values can be derived: 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.

[citation needed] 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] Tetration can be extended to infinite heights; i.e., for certain a and n values in

, there exists a well defined result for an infinite n. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity.

The trend towards 2 can be seen by evaluating a small finite tower: In general, the infinitely iterated exponential

as n goes to infinity, converges for e−e ≤ x ≤ e1/e, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler.

As the limit y = ∞x (if existent on the positive real line, i.e. for e−e ≤ x ≤ e1/e) must satisfy xy = y we see that x ↦ y = ∞x is (the lower branch of) the inverse function of y ↦ x = y1/y.

is consistent with the rule because 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.

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

[1][14] In 2017, it was proven[15] that there exists a unique function F which is a solution of the equation F(z + 1) = exp(F(z)) and satisfies the additional conditions that F(0) = 1 and F(z) approaches the fixed points of the logarithm (roughly 0.318 ± 1.337i) as z approaches ±i∞ and that F is holomorphic in the whole complex z-plane, except the part of the real axis at z ≤ −2.

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 function S satisfies the tetration equations S(z + 1) = exp(S(z)), S(0) = 1, and if αn and βn approach 0 fast enough it will be analytic on a neighborhood of the positive real axis.

However, if some elements of {α} or {β} are not zero, then function S has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients {α} and {β} are, the further away these singularities are from the real axis.

One can prove by induction that for every elementary recursive function f, there is a constant c such that We denote the right hand side by

it does not have any real square super-roots, but the formula given above yields countably infinitely many complex ones for any finite x not equal to 1.

[citation needed] Just as with the extension of tetration to infinite heights, the super-root can be extended to n = ∞, being well-defined if 1/e ≤ x ≤ e. Note that

Once a continuous increasing (in x) definition of tetration, xa, is selected, the corresponding super-logarithm

The function sloga x satisfies: 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: For each graph H on h vertices and each ε > 0, define Then each graph G on n vertices with at most nh/D copies of H can be made H-free by removing at most εn2 edges.