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In mathematics, tetration (or hyper4) is the next hyper operator after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra (four) and iteration. Tetration is used for the notation of very large numbers. Shown here are examples of the first four hyper operators, with tetration as the fourth (and succession, the unary operation denoted taking and yielding the number after , as the 0th):
where each operation is defined by iterating the previous one (the next operation in the sequence is pentation). The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of n, while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.
Addition () is the most basic operation, multiplication () is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a, and exponentiation () can be thought of as a chained multiplication involving n numbers a. Analogously, tetration () can be thought of as a chained power involving n numbers a. The parameter a may be called the baseparameter in the following, while the parameter n in the following may be called the heightparameter (which is integral in the first approach but may be generalized to fractional, real and complex heights, see below).
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For any positive real and nonnegative integer , we define by:
As we can see from the definition, when evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:
Note that exponentiation is not associative, so evaluating the expression in the other order will lead to a different answer:
Thus, the exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as rightassociative.
When a and 10 are coprime, we can compute the last m decimal digits of using Euler's theorem.
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 counterrationale.
Tetration is often confused with closely related functions and expressions. This is because much of the terminology that is used with them can be used with tetration. Here are a few related terms:
Form  Terminology 

Tetration  
Iterated exponentials  
Nested exponentials (also towers)  
Infinite exponentials (also towers) 
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 of these styles can be used for higher iterations as well (hyper5, hyper6, and so on).^{[clarification needed]}
Name  Form  Description 

Standard notation  Used by Maurer [1901] and Goodstein [1947]; Rudy Rucker's book Infinity and the Mind popularized the notation.  
Knuth's uparrow notation  Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.  
Conway chained arrow notation  Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain  
Ackermann function  Allows the special case to be written in terms of the Ackermann function.  
Iterated exponential notation  Allows simple extension to iterated exponentials from initial values other than 1.  
Hooshmand notation^{[5]}  
Hyper operator notation  Allows extension by increasing the number 4; this gives the family of hyper operations  
ASCII notation  a^^n 
Since the uparrow is used identically to the caret (^ ), the tetration operator may be written as (^^ ). 
One notation above uses iterated exponential notation; in general this is defined as follows:
There are not as many notations for iterated exponentials, but here are a few:
Name  Form  Description 

Standard notation  Euler coined the notation , and iteration notation has been around about as long.  
Knuth's uparrow notation  Allows for superpowers and superexponential function by increasing the number of arrows; used in the article on large numbers.  
Ioannis Galidakis' notation  Allows for large expressions in the base.^{[6]}  
ASCII (auxiliary)  a^^n@x 
Based on the view that an iterated exponential is auxiliary tetration. 
ASCII (standard)  exp_a^n(x) 
Based on standard notation. 
J Notation  x^^:(n1)x 
Repeats the exponentiation. See J (programming language)^{[7]} 
In the following table, most values are too large to write in scientific notation, so iterated exponential notation is employed to express them in base 10. The values containing a decimal point are approximate.
1  1  1  1 
2  4  16  65,536 
3  27  7,625,597,484,987  
4  256  
5  3,125  
6  46,656  
7  823,543  
8  16,777,216  
9  387,420,489  
10  10,000,000,000 
Tetration can be extended to define and other domains as well.
The exponential is non consistently defined. Thus, the tetrations are not clearly defined by the formula given earlier. However, is well defined, and exists:
Thus we could consistently define . This is equivalent to defining .
Under this extension, , so the rule from the original definition still holds.
Since complex numbers can be raised to powers, tetration can be applied to bases of the form , where is the square root of −1. For example, where , tetration is achieved by using the principal branch of the natural logarithm, and using Euler's formula we get the relation:
This suggests a recursive definition for given any :
The following approximate values can be derived:
Approximate Value  

i  
Solving the inverse relation as in the previous section, yields the expected and , with negative values of n giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit , 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 power tower 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.
Tetration can be extended to infinite heights (n in ). This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity. For example, converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:
In general, the infinite power tower , defined as the limit of as n goes to infinity, converges for e^{−e} ≤ x ≤ e^{1/e}, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler. The limit, should it exist, is a positive real solution of the equation y = x^{y}. Thus, x = y^{1/y}. The limit defining the infinite tetration of x fails to converge for x > e^{1/e} because the maximum of y^{1/y} is e^{1/e}.
This may be extended to complex numbers z with the definition:
where W(z) represents Lambert's W function.
As the limit y = ^{∞}x (if existent, i.e. for e^{−e} < x < e^{1/e}) must satisfy x^{y} = y we see that x ↦ y = ^{∞}x is (the lower branch of) the inverse function of y ↦ x = y^{1/y}.
In order to preserve the original rule:
for negative values of we must use the recursive relation:
Thus:
However smaller negative values cannot be well defined in this way because
which is not well defined.
Note further that for any definition of 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 . Various approaches are mentioned below.
In general the problem is finding, for any real a > 0, a superexponential function over real x > 2 that satisfies
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.
Fortunately, any solution that satisfies one of these in an interval of length one can be extended to a solution for all positive real numbers. When is defined for an interval of length one, the whole function easily follows for all x > 2.
A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
hence:
Approximation  Domain 

for  
for  
for 
and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by .
A main theorem in Hooshmand's paper^{[5]} states: Let . If is continuous and satisfies the conditions:
then is uniquely determined through the equation
where denotes the fractional part of x and is the iterated function of the function .
The proof is that the second through fourth conditions trivially imply that f is a linear function on [1, 0].
The linear approximation to natural tetration function is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:
If is a continuous function that satisfies:
then . [Here is Hooshmand's name for the linear approximation to the natural tetration function.]
The proof is much the same as before; the recursion equation ensures that and then the convexity condition implies that is linear on (1, 0).
Therefore the linear approximation to natural tetration is the only solution of the equation and which is convex on . All other sufficientlydifferentiable solutions must have an inflection point on the interval (1, 0).
A quadratic approximation (to the differentiability requirement) is given by:
which is differentiable for all , but not twice differentiable. If this is the same as the linear approximation.
A cubic approximation and a method for generalizing to approximations of degree n are given at.^{[8]}
There is a conjecture^{[9]} 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 0.31813150520476413531 ± 1.33723570143068940890i) as z approaches ±i∞ and that F is holomorphic in the whole complex zplane, except the part of the real axis at z≤−2. This function is shown in the figure at right. The complex double precision approximation of this function is available online.^{[10]}
The requirement of holomorphism of tetration is important for the uniqueness. Many functions can be constructed as
where and are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of .
The function S satisfies the tetration equations S(z+1)=exp(S(z)), S(0)=1, and if α_{n} and β_{n} 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.
The extension of tetration into the complex plane is thus essential for the uniqueness; the realanalytic tetration is not unique.
It is not known if ^{n}π or ^{n}e is an integer for any positive integer n.
Exponentiation has two inverse relations; roots and logarithms. Analogously, the inverse relations of tetration are often called the superroot, and the superlogarithm.
The superroot is the inverse relation of tetration with respect to the base: if , then y is an nth super root of x. For example,
so 2 is the 4th superroot of 65,536 and
so 3 is the 3rd superroot (or super cube root) of 7,625,597,484,987.
The 2ndorder superroot, square superroot, or super square root has two equivalent notations, and . It is the inverse of and can be represented with the Lambert W function:^{[11]}
The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when :
Like square roots, the square superroot of x may not have a single solution. Unlike square roots, determining the number of square superroots of x may be difficult. In general, if , then x has two positive square superroots between 0 and 1; and if , then x has one positive square superroot greater than 1. If x is positive and less than it doesn't have any real square superroots, but the formula given above yields countably infinitely many complex ones for any finite x not equal to 1.^{[11]} The function has been used to determine the size of data clusters.^{[12]}
For each integer n > 2, the function ^{n}x is defined and increasing for x ≥ 1, and ^{n}1 = 1, so that the nth superroot of x, , exists for x ≥ 1.
However, if the linear approximation above is used, then if 1 < y ≤ 0, so cannot exist.
Other superroots are expressible under the same basis^{[clarification needed]} used with normal roots: super cube roots, the function^{[dubious – discuss]} that produces y when , can be expressed as ; the 4th superroot can be expressed as ; and it can therefore be said that the n^{th} superroot is . Note that may not be uniquely defined, because there may be more than one n^{th} root. For example, x has a single (real) superroot if n is odd, and up to two if n is even.^{[citation needed]}
The superroot can be extended to , and this shows a link to the mathematical constant e as it is only welldefined if 1/e ≤ x ≤ e (see extension of tetration to infinite heights). Note that implies that and thus that . Therefore, when it is well defined, and thus it is an elementary function. For example, .
Once a continuous increasing (in x) definition of tetration, ^{x}a, is selected, the corresponding superlogarithm slog_{a} x is defined for all real numbers x, and a > 1.
The function satisfies:
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