In mathematics, tetration (or hyper4) is iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra (four) and iteration. Tetration is used for the notation of very large numbers. The notation means , which is the application of exponentiation times.
The first four hyperoperations are shown here, with tetration being the fourth of these (in this case, the unary operation succession, , is considered to be the zeroth operation).
Here, succession (a' = a + 1) is the most basic operation; addition (a + n) is a primary operation, though for natural numbers it can be thought of as a chained succession of n successors of a; multiplication () is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a. Exponentiation () can be thought of as a chained multiplication involving n numbers a, and analogously, tetration () can be thought of as a chained power involving n numbers a. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.
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). Tetration is read as "the nth tetration of a".
For any positive real and nonnegative integer , we can define recursively as:
This formal definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to other heights such as , , and .
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.
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:
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 notations can also be used to describe other hyperoperations, while some are limited to tetration and have no immediate extension.
Name  Form  Description 

Rudy Rucker 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 notations^{[5]}  Used by M. H. Hooshmand [2006].  
Hyperoperation notations  Allows extension by increasing the number 4; this gives the family of hyperoperations.  
Double caret notation  a^^n

Since the uparrow is used identically to the caret (^ ), tetration may be written as (^^ ); convenient for ASCII.

Bowers's operators 

Similar to the hyperoperation notations. 
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.  
Text notation  exp_a^n(x)

Based on standard notation; convenient for ASCII. 
J Notation  x^^:(n1)x

Repeats the exponentiation. See J (programming language)^{[6]} 
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.
1  1  1  1  1 
2  4  16  65,536  2^{65,536} or (2.00353 × 10^{19,728}) 
3  27  7,625,597,484,987  (3.6 × 10^{12} digits)  
4  256  1.34078 × 10^{154}  (8.1 × 10^{153} digits)  
5  3,125  1.91101 × 10^{2,184}  (1.3 × 10^{2,184} digits)  
6  46,656  2.65912 × 10^{36,305}  (2.1 × 10^{36,305} digits)  
7  823,543  3.75982 × 10^{695,974}  (3.2 × 10^{695,974} digits)  
8  16,777,216  6.01452 × 10^{15,151,335}  (5.4 × 10^{15,151,335} digits)  
9  387,420,489  4.28125 × 10^{369,693,099}  (4.1 × 10^{369,693,099} digits)  
10  10,000,000,000  10^{10,000,000,000}  (10^{1010} digits) 
Tetration has several properties that are similar to exponentiation, as well as properties that are specific to the operation and are lost or gained from exponentiation. Because exponentiation does not commute, the product and power rules do not have an analogue with tetration; the statements and are not necessarily true for all cases.^{[7]}
However, tetration does follow a different property, in which . This fact is most clearly shown using the recursive definition. From this property, a proof follows that , which allows for switching b and c in certain equations. The proof goes as follows:
When a number x and 10 are coprime, it is possible to compute the last m decimal digits of using Euler's theorem, for any integer m.
When evaluating tetration expressed as an "exponentiation tower", the 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:
Evaluating the expression the left to right is considered less interesting; evaluating left to right, any expression can be simplified to be .^{[8]} Because of this, the towers must be evaluated from right to left (or top to bottom). Computer programmers refer to this choice as rightassociative.
Tetration can be extended in two different ways; in the equation , both the base a and the height n can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the positive integers to different domains, including , complex functions such as , and heights of infinite , the more limited properties of tetration reduce the ability to extend tetration.
The exponential is not 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 z = a + bi (where a and b are real). For example, in ^{n}z 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+1}i = a' + b'i given any ^{n}i = a + bi:
The following approximate values can be derived:
Approximate value  

i  
0.2079  
0.9472 + 0.3208i  
0.0501 + 0.6021i  
0.3872 + 0.0305i  
0.7823 + 0.5446i  
0.1426 + 0.4005i  
0.5198 + 0.1184i  
0.5686 + 0.6051i 
Solving the inverse relation, as in the previous section, yields the expected ^{0}i = 1 and ^{−1}i = 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.
Tetration can be extended to infinite heights;^{[9]} 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. 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 infinitely iterated exponential , 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.^{[10]} 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 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}.
We can use the recursive rule for tetration,
to prove :
Substituting −1 for k gives
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.^{[8]}
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 . There have, however, been multiple approaches towards the issue, and different approaches are outlined below.
In general the problem is finding, for any real a > 0, a superexponential function over real x > −2 that satisfies
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.
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 −1 < x < 0  
for 0 < x < 1  
for 1 < x < 2 
and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by . It is continuously differentiable for if and only if . For example, using these methods and
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).
Beyond linear approximations, a quadratic approximation (to the differentiability requirement) is given by:
which is differentiable for all , but not twice differentiable. For example, If this is the same as the linear approximation.^{[12]}
Note that because of the way it is calculated, this function does not "cancel out", contrary to exponents, where . Namely,
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.^{[12]}^{[13]}
It has now been proven^{[14]} 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 zplane, except the part of the real axis at z ≤ −2. This proof confirms a previous conjecture.^{[15]} The complex map of this function is shown in the figure at right. The proof also works for other bases besides e, as long as the base is bigger than . The complex double precision approximation of this function is available online.
The requirement of the tetration being holomorphic is important for its 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.
Tetration (restricted to ) is not an elementary recursive function. 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 . Suppose on the contrary that tetration is elementary recursive. is also elementary recursive. By the above inequality, there is a constant c such that . By letting , we have that , a contradiction.
Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the superroot, and the superlogarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function , the two inverses are the cube superroot of y and the super logarithm base y of x.
The superroot is the inverse operation of tetration with respect to the base: if , then y is an nth super root of x ( or ).
For example,
so 2 is the 4th superroot of 65,536.
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:^{[16]}
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.^{[16]} The function has been used to determine the size of data clusters.^{[17]}
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.
In the same way as the square superroot, terminology for other super roots can be based on the normal roots: "cube superroots" can be expressed as ; the "4th superroot" can be expressed as ; and 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.
Just as with the extension of tetration to infinite heights, the superroot can be extended to , being welldefined if 1/e ≤ x ≤ e. Note that and thus that . Therefore, when it is well defined, and, unlike normal tetration, is an elementary function. For example, .
It follows from the Gelfond–Schneider theorem that superroot for any positive integer n is either integer or transcendental, and is either integer or irrational.^{[18]} It is still an open question whether irrational superroots are transcendental in the latter case.
Once a continuous increasing (in x) definition of tetration, ^{x}a, is selected, the corresponding superlogarithm or is defined for all real numbers x, and a > 1.
The function slog_{a} 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:
A double exponential function is a constant raised to the power of an exponential function. The general formula is , which grows much more quickly than an exponential function. For example, if a = b = 10:
Factorials grow more quickly than exponential functions, but much more slowly than doubly exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions.
The inverse of the double exponential function is the double logarithm ln(ln(x)).
ELEMENTARYIn computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes
The name was coined by László Kalmár, in the context of recursive functions and undecidability; most problems in it are far from elementary. Some natural recursive problems lie outside ELEMENTARY, and are thus NONELEMENTARY. Most notably, there are primitive recursive problems that are not in ELEMENTARY. We know
Whereas ELEMENTARY contains bounded applications of exponentiation (for example, ), PR allows more general hyper operators (for example, tetration) which are not contained in ELEMENTARY.
Exponential factorialThe exponential factorial of a positive integer n, denoted by n$, is n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on and so forth, that is,
The exponential factorial can also be defined with the recurrence relation
The first few exponential factorials are 1, 1, 2, 9, 262144, etc. (sequence A049384 in the OEIS). So, for example, 262144 is an exponential factorial since
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The exponential factorial of 5 is 5^{262144} which is approximately 6.206069878660874 × 10^{183230}.
The sum of the reciprocals of the exponential factorials from 1 onwards is the transcendental number
This sum is transcendental because it is a Liouville number.
Like tetration, there is currently no accepted method of extension of the exponential factorial function to real and complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function.
GoogolplexA googolplex is the number 10googol, or equivalently, 10(10100). Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes, that is, a 1 followed by a googol zeroes.
Hereditarily finite setIn mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.
HyperoperationIn mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).
After that, the sequence proceeds with further binary operations extending beyond exponentiation, using rightassociativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written as using n − 2 arrows in Knuth's uparrow notation. Each hyperoperation may be understood recursively in terms of the previous one by:
It may also be defined according to the recursion rule part of the definition, as in Knuth's uparrow version of the Ackermann function:
This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes' number and googolplexplex (e.g. is much larger than Skewes’ number and googolplexplex), but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3).
This recursion rule is common to many variants of hyperoperations (see below in definition).
Iterated functionIn mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial number, the result of applying a given function is fed again in the function as input, and this process is repeated.
Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.
Iterated logarithmIn computer science, the iterated logarithm of , written log* (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to . The simplest formal definition is the result of this recurrence relation:
On the positive real numbers, the continuous superlogarithm (inverse tetration) is essentially equivalent:
but on the negative real numbers, logstar is , whereas for positive , so the two functions differ for negative arguments.
The iterated logarithm accepts any positive real number and yields an integer. Graphically, it can be understood as the number of "zigzags" needed in Figure 1 to reach the interval on the xaxis.
In computer science, lg* is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base ) instead of the natural logarithm (with base e).
Mathematically, the iterated logarithm is welldefined for any base greater than , not only for base and base e.
Knuth's uparrow notationIn mathematics, Knuth's uparrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated addition and exponentiation as iterated multiplication. Continuing in this manner leads to tetration (iterated exponentiation) and to the remainder of the hyperoperation sequence, which is commonly denoted using Knuth arrow notation. This notation allows for a simple description of numbers far larger than can be explicitly written out.
A single arrow means exponentiation (iterated multiplication); more than one arrow means iterating the operation associated with one fewer arrow.
For example:
The general definition of the notation (by recursion) is as follows (for integer a and nonnegative integers b and n):
Here, ↑^{n} stands for n arrows, so for example
Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive integers, or more generally, large positive real numbers, but it may also be used in other contexts.
Very large numbers often occur in fields such as mathematics, cosmology, cryptography, and statistical mechanics. Sometimes people refer to numbers as being "astronomically large". However, it is easy to mathematically define numbers that are much larger even than those used in astronomy.
List of exponential topicsThis is a list of exponential topics, by Wikipedia page. See also list of logarithm topics.
Accelerating change
Approximating natural exponents (log base e)
Artin–Hasse exponential
Bacterial growth
Baker–Campbell–Hausdorff formula
Cell growth
Barometric formula
Beer–Lambert law
Characterizations of the exponential function
Catenary
Compound interest
De Moivre's formula
Derivative of the exponential map
DoléansDade exponential
Doubling time
efolding
Elimination halflife
Error exponent
Euler's formula
Euler's identity
e (mathematical constant)
Exponent
Exponent bias
Exponential (disambiguation)
Exponential backoff
Exponential decay
Exponential dichotomy
Exponential discounting
Exponential diophantine equation
Exponential dispersion model
Exponential distribution
Exponential error
Exponential factorial
Exponential family
Exponential field
Exponential formula
Exponential function
Exponential generating function
ExponentialGolomb coding
Exponential growth
Exponential hierarchy
Exponential integral
Exponential integrator
Exponential map (Lie theory)
Exponential map (Riemannian geometry)
Exponential notation
Exponential object (category theory)
Exponential polynomials—see also Touchard polynomials (combinatorics)
Exponential response formula
Exponential sheaf sequence
Exponential smoothing
Exponential stability
Exponential sum
Exponential time
Subexponential time
Exponential tree
Exponential type
Exponentially equivalent measures
Exponentiating by squaring
Exponentiation
Fermat's Last Theorem
Forgetting curve
Gaussian function
Gudermannian function
Halfexponential function
Halflife
Hyperbolic function
Inflation, inflation rate
Interest
Lifetime (physics)
Limiting factor
Lindemann–Weierstrass theorem
List of integrals of exponential functions
List of integrals of hyperbolic functions
Lyapunov exponent
Malthusian catastrophe
Malthusian growth model
Marshall–Olkin exponential distribution
Matrix exponential
Moore's law
Nachbin's theorem
Piano key frequencies
padic exponential function
Power law
Proof that e is irrational
Proof that e is transcendental
Qexponential
Radioactive decay
Rule of 70, Rule of 72
Scientific notation
Six exponentials theorem
Spontaneous emission
Superexponentiation
Tetration
Versor
Wilkie's theorem
Zenzizenzizenzic
PR (complexity)PR is the complexity class of all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided by such a function. This includes addition, multiplication, exponentiation, tetration, etc.
The Ackermann function is an example of a function that is not primitive recursive, showing that PR is strictly contained in R (Cooper 2004:88).
On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitiverecursive function in the following sense: given an input (M, k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing. Then the union of the outputs, over all possible inputs (M, k), is exactly the set of M that halt.
PR strictly contains ELEMENTARY.
PR does not contain "PRcomplete" problems (assuming, e.g., reductions that belong to ELEMENTARY). In practice, many problems that are not in PR but just beyond are complete (Schmitz 2016).
PentationIn mathematics, pentation is the next hyperoperation after tetration but before hexation. It is defined as iterated (repeated) tetration, just as tetration is iterated exponentiation. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b times. For instance, using hyperoperation notation for pentation and tetration, means tetrating 2 to itself 3 times, or . This can then be reduced to
Power towerPower tower may refer to:
Power Tower, a thrill ride at Valleyfair and Cedar Point amusement parks in the USA
Power tower (power takeoff), a type of mechanical power takeoff (PTO)
Solar power tower, a type of solar power plant
Tetration, a mathematical operation also known as power tower, hyperpower, or superexponentiation
Transmission tower, usually a tall steel lattice tower supporting an overhead electric power line (aka electricity pylon, hydro tower, etc.)
Power tower (exercise), equipment can include pull up handles, dip bars, push up grips and captain's chair forearm pads
Power tower (Linz), an office building in Linz, Austria
Successor functionIn mathematics, the successor function or successor operation is a primitive recursive function S such that S(n) = n+1 for each natural number n.
For example, S(1) = 2 and S(2) = 3. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.
SuperlogarithmIn mathematics, the superlogarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, superroots and superlogarithms. There are several ways of interpreting superlogarithms:
For positive integer values, the superlogarithm with basee is equivalent to the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm). However, this is not true for negative values and so cannot be considered a full definition. The precise definition of the superlogarithm depends on a precise definition of nonintegral tetration (that is, for y not an integer). There is no clear consensus on the definition of nonintegral tetration and so there is likewise no clear consensus on the superlogarithm for noninteger range.
SuperfunctionIn mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression
Then, S(z;x) can be interpreted as the superfunction of the function f(x). Such a definition is valid only for a positive integer index z. The variable x is often omitted. Much study and many applications of superfunctions employ various extensions of these superfunctions to complex and continuous indices; and the analysis of the existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of superfunctions.
Superpower (disambiguation)Superpower, a state with the ability to influence events and project power on a worldwide scale
Second Superpower, a term used to conceptualize a global civil society as a counterpoint to the United States of America
Energy superpower
Potential superpowersSuperpower may also refer to:
Superpower (ability), extraordinary powers mostly possessed by fictional characters, commonly in American comic books
Tetration, superpower or hyperpower is used as synonym of tetration.
^^^^ may refer to:
A kaomoji
Tetration, the ASCII form of the tetration operator
Record separator, control character in the caret notation
Logical exclusive or, operator in the GLSL language
Primary  

Inverse for left argument  
Inverse for right argument  
Related articles 
Examples in numerical order  

Expression methods 
 
Related articles (alphabetical order)  
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