Editing Tetration

{{Short description|Arithmetic operation}}
{{Distinguish|Titration (disambiguation){{!}}Titration}}
{{for|repeated tetration|Pentation}}
{{Use dmy dates|date=August 2020|cs1-dates=y}}
[[File:TetrationComplexColor.png|thumb|alt=A colorful graphic with brightly colored loops that grow in intensity as the eye goes to the right|[[Domain coloring]] of the [[holomorphic function|holomorphic]] tetration <math>{}^{z}e</math>, with [[hue]] representing the function [[Argument (complex analysis)|argument]] and [[brightness]] representing magnitude]]
[[File:TetrationConvergence2D.svg|thumb|alt=A line graph with curves that bend upward dramatically as the values on the x-axis get larger|<math>{}^{n}x</math>, for {{math|1=''n'' = 2, 3, 4, ...}}, showing convergence to the infinitely iterated exponential between the two dots]]

In [[mathematics]], '''tetration''' (or '''hyper-4''') is an [[operation (mathematics)|operation]] based on [[iterated]], or repeated, [[exponentiation]]. There is no standard [[mathematical notation|notation]] for tetration, though [[Knuth's up arrow notation]] <math>\uparrow \uparrow</math> and the left-exponent <math>{}^{x}b</math> are common.

Under the definition as repeated exponentiation, <math>{^{n}a}</math> means <math>{a^{a^{\cdot^{\cdot^{a}}}}}</math>, where ''{{mvar|n}}'' copies of ''{{mvar|a}}'' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation <math>n-1</math> times. ''{{mvar|n}}'' is called the "height" of the function, while ''{{mvar|a}}'' is called the "base," analogous to exponentiation. It would be read as "the {{mvar|n}}th tetration of {{mvar|a}}". For example, 2 tetrated to 4 (or the fourth tetration of 2) is <math>{^{4}2}=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536</math>.

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

Tetration is also defined recursively as
: <math>{a \uparrow \uparrow n} := \begin{cases} 1 &\text{if }n=0, \\ a^{a \uparrow\uparrow (n-1)} &\text{if }n>0, \end{cases}</math>
allowing for the [[holomorphic function|holomorphic]] extension of tetration to [[Natural numbers|non-natural numbers]] such as [[Real number|real]], [[Complex number|complex]], and [[ordinal number]]s, which was proved in 2017.

The two inverses of tetration are called '''[[#Super-root|super-root]]''' and '''[[#Super-logarithm|super-logarithm]]''', analogous to the [[nth root]] and the logarithmic functions. None of the three functions are [[elementary function|elementary]].

Tetration is used for the [[Large numbers#Standardized system of writing|notation of very large numbers]].

== Introduction ==
The first four [[hyperoperation]]s are shown here, with tetration being considered the fourth in the series. The [[unary operation]] [[Successor function|succession]], defined as <math>a' = a + 1</math>, is considered to be the zeroth operation.

#[[Addition]] <math display="block">a + n = a + \underbrace{1 + 1 + \cdots + 1}_n</math> {{mvar|n}} copies of 1 added to {{mvar|a}} combined by succession.
#[[Multiplication]] <math display="block">a \times n = \underbrace{a + a + \cdots + a}_n</math> {{mvar|n}} copies of {{mvar|a}} combined by addition.
#[[Exponentiation]] <math display="block">a^n = \underbrace{a \times a \times \cdots \times a}_n</math> {{mvar|n}} copies of {{mvar|a}} combined by multiplication.
#Tetration <math display="block">{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n</math> {{mvar|n}} copies of {{mvar|a}} combined by exponentiation, right-to-left.

Importantly, nested exponents are interpreted from the top down: {{tmath|a^{b^c} }} means {{tmath|a^{\left(b^c \right)} }} and not {{tmath|\left(a^b \right)^c.}}

Succession, <math>a_{n+1} = a_n + 1</math>, is the most basic operation; while addition (<math>a + n</math>) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of <math>n</math> successors of <math>a</math>; multiplication (<math>a \times n</math>) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving <math>n</math> numbers of <math>a</math>. Exponentiation can be thought of as a chained multiplication involving <math>n</math> numbers of <math>a</math> and tetration (<math>^{n}a</math>) as a chained power involving <math>n</math> numbers <math>a</math>. Each of the operations above are defined by iterating the previous one;<ref name="uwu">Neyrinck, Mark. [http://skysrv.pha.jhu.edu/~neyrinck/extessay.pdf An Investigation of Arithmetic Operations.] Retrieved 9 January 2019.</ref> however, unlike the operations before it, tetration is not an [[elementary function]].

The parameter <math>a</math> is referred to as the '''base''', while the parameter <math>n</math> may be referred to as the '''height'''. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive [[real number|real]] <math>a > 0</math> and non-negative [[integer]] <math>n \ge 0</math>, we can define <math>\,\! {^{n}a}</math> recursively as:<ref name="uwu" />
: <math>{^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left(^{(n-1)}a\right)} &\text{if }n>0 \end{cases}</math>
The recursive definition is equivalent to repeated exponentiation for [[Natural number|natural]] heights; however, this definition allows for extensions to the other heights such as <math>^{0}a</math>, <math>^{-1}a</math>, and <math>^{i}a</math> as well – many of these extensions are areas of active research.

== Terminology ==
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.

* The term ''tetration'', introduced by Goodstein in his 1947 paper ''Transfinite Ordinals in Recursive Number Theory''<ref>{{cite journal |author=R. L. Goodstein |title=Transfinite ordinals in recursive number theory |jstor=2266486 |journal=Journal of Symbolic Logic |volume=12 |year=1947 |issue=4 |doi=10.2307/2266486 |pages=123–129|s2cid=1318943 }}</ref> (generalizing the recursive base-representation used in [[Goodstein's theorem]] to use higher operations), has gained dominance. It was also popularized in [[Rudy Rucker]]'s ''[[Infinity and the Mind]]''.
* The term ''superexponentiation'' was published by Bromer in his paper ''Superexponentiation'' in 1987.<ref>{{cite journal |author=N. Bromer |title=Superexponentiation |journal=Mathematics Magazine |volume=60 |issue=3 |year=1987 |pages=169–174 |jstor=2689566|doi=10.1080/0025570X.1987.11977296 }}</ref> It was used earlier by Ed Nelson in his book Predicative Arithmetic, [[Princeton University Press]], 1986.
* The term ''hyperpower''<ref>{{cite journal |author=J. F. MacDonnell |title=Somecritical points of the hyperpower function <math>x^{x^{\dots}}</math> |journal=International Journal of Mathematical Education |year=1989 |volume=20 |issue=2 |pages=297–305 |mr=994348 |url=http://www.faculty.fairfield.edu/jmac/ther/tower.htm |doi=10.1080/0020739890200210|url-access=subscription }}</ref> is a natural combination of ''hyper'' and ''power'', which aptly describes tetration. The problem lies in the meaning of ''hyper'' with respect to the [[hyperoperation]] sequence. When considering hyperoperations, the term ''hyper'' refers to all ranks, and the term ''super'' refers to rank 4, or tetration. So under these considerations ''hyperpower'' is misleading, since it is only referring to tetration.
* The term ''power tower''<ref>{{MathWorld |urlname=PowerTower |title=Power Tower}}</ref> is occasionally used, in the form "the power tower of order {{mvar|n}}" for <math>{\ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop n}</math>. Exponentiation is easily misconstrued: note that the operation of raising to a power is right-associative (see [[#Direction of evaluation|below]]). Tetration is iterated ''exponentiation'' (call this [[Operator associativity#rightAssociative|right-associative]] operation ^), starting from the top right side of the expression with an instance a^a (call this value c). Exponentiating the next leftward a (call this the 'next base' b), is to work leftward after obtaining the new value b^c. Working to the left, use the next a to the left, as the base b, and evaluate the new b^c. 'Descend down the tower' in turn, with the new value for c on the next downward step.

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

{|class="wikitable"
|+Terms related to tetration
!scope="col"| Terminology
!scope="col"| Form
|-
!scope="row"|Tetration
|<math>a^{a^{\cdot^{\cdot^{a^a}}}}</math>
|-
!scope="row"|Iterated exponentials
|<math>a^{a^{\cdot^{\cdot^{a^x}}}}</math>
|-
!scope="row"|Nested exponentials (also towers)
|<math>a_1^{a_2^{\cdot^{\cdot^{a_n}}}}</math>
|-
!scope="row"|Infinite exponentials (also towers)
|<math>a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}</math>
|}

In the first two expressions {{mvar|a}} is the ''base'', and the number of times {{mvar|a}} appears is the ''height'' (add one for {{mvar|x}}). In the third expression, {{mvar|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 function|iterated]] [[power (mathematics)|powers]] or iterated [[exponential function|exponentials]].

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

{|class="wikitable"
|+Notation styles for tetration
!scope="col"| Name
!scope="col"| Form
!scope="col"| Description
|-
! scope="row" | [[Knuth's up-arrow notation]]
| <math>\begin{align}
a {\uparrow\uparrow} n \\
a {\uparrow}^2 n
\end{align}</math>
| Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
|-
! scope="row" | [[Conway chained arrow notation]]
| <math>a \rightarrow n \rightarrow 2</math>
| Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain.
|-
! scope="row" | [[Ackermann function]]
| <math>{}^{n}2 = \operatorname{A}(4, n - 3) + 3</math>
| Allows the special case <math>a=2</math> to be written in terms of the Ackermann function.
|-
! scope="row" | Iterated exponential notation
| <math>\exp_a^n(1)</math>
| Allows simple extension to iterated exponentials from initial values other than 1.
|-
! scope="row" | Hooshmand notations<ref name="uxp">{{cite journal |author-first=M. H. |author-last=Hooshmand |date=2006 |title=Ultra power and ultra exponential functions |journal=[[Integral Transforms and Special Functions]] |volume=17 |issue=8 |pages=549–558 |doi=10.1080/10652460500422247 |s2cid=120431576}}</ref>
| <math>\begin{align}
  &\operatorname{uxp}_a n \\[2pt]
  &a^{\frac{n}{}}
\end{align}</math>
| Used by M. H. Hooshmand [2006].
|-
! scope="row" | [[Hyperoperation]] notations
| <math>\begin{align}
  &a [4] n \\[2pt]
  &H_4(a, n)
\end{align}</math>
| Allows extension by increasing the number 4; this gives the family of [[hyperoperation]]s.
|-
!scope="row"| Double caret notation
| {{code|a^^n}}
| Since the up-arrow is used identically to the caret (<code>^</code>), tetration may be written as (<code>^^</code>); convenient for [[ASCII]].
|}

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

: <math>\exp_a^n(x) = a^{a^{\cdot^{\cdot^{a^x}}}}</math> with {{mvar|n}} {{mvar|a}}s.

There are not as many notations for iterated exponentials, but here are a few:
{| class="wikitable"
|+Notation styles for iterated exponentials
!scope="col"| Name
!scope="col"| Form
!scope="col"| Description
|-
!scope="row"| Standard notation
| <math>\exp_a^n(x)</math>
| [[Leonhard Euler|Euler]] coined the notation <math>\exp_a(x) = a^x</math>, and iteration notation <math>f^n(x)</math> has been around about as long.
|-
!scope="row"| Knuth's up-arrow notation
| <math>(a{\uparrow}^2(x))</math>
| Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on [[large numbers]].
|-
!scope="row"| Text notation
| {{code|2=tex|exp_a^n(x)}}
| Based on standard notation; convenient for [[ASCII]].
|-
!scope="row"| J Notation
| {{code|2=j|x^^:(n-1)x}}
| Repeats the exponentiation. See [[J (programming language)]]<ref>{{cite web |title=Power Verb |url=http://www.jsoftware.com/help/dictionary/d202n.htm |work=J Vocabulary |publisher=J Software |access-date=28 October 2011}}</ref>
|-
!scope="row"| Infinity barrier notation
| <math>a\uparrow\uparrow n|x</math>
| Jonathan Bowers coined this,<ref>{{cite web |title=Spaces |url=http://www.polytope.net/hedrondude/spaces.htm |access-date=17 February 2022}}</ref> and it can be extended to higher hyper-operations
|}

== Examples ==
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. Usually, the limit that can be calculated in a numerical calculation program such as [[Wolfram Alpha]] is 3↑↑4, and the number of digits up to 3↑↑5 can be expressed.

{| class="wikitable"
|+Examples of tetration
!scope="col"| <math>x</math>
!scope="col"| <math>{}^{2}x</math>
!scope="col"| <math>{}^{3}x</math>
!scope="col"| <math>{}^{4}x</math>
!scope="col"| <math>{}^{5}x</math>
!scope="col"| <math>{}^{6}x</math>
!scope="col"| <math>{}^{7}x</math>
|- align=right
!scope="row"| 1
| 1
| 1
| 1
| 1
| 1
| 1
|- align=right
!scope="row"| 2
| 4 (2{{sup|2}})
| 16 (2{{sup|4}})
| 65,536 (2{{sup|16}})
| 2.00353 × 10{{sup|19,728}}
| <math>\exp_{10}^3(4.29508)</math> (10{{sup|6.03123×10{{sup|19,727}}}})
| <math>\exp_{10}^4(4.29508)</math>
|- align=right
!scope="row"| 3
| 27 (3{{sup|3}})
| 7,625,597,484,987 (3{{sup|27}})
| 1.25801 × 10{{sup|3,638,334,640,024}} <ref name="tdm">DiModica, Thomas. [https://github.com/TediusTimmy/CiteMyself/blob/trunk/Tetration/README.md Tetration Values.] Retrieved 15 October 2023.</ref>
| <math>\exp_{10}^4(1.09902)</math>
(10{{sup|6.00225×10{{sup|3,638,334,640,023}}}})
| <math>\exp_{10}^5(1.09902)</math>
| <math>\exp_{10}^6(1.09902)</math>
|- align=right
!scope="row"| 4
| 256 (4{{sup|4}})
| 1.34078 × 10{{sup|154}} (4{{sup|256}})
| <math>\exp_{10}^3(2.18726)</math> (10{{sup|8.0723×10{{sup|153}}}})
| <math>\exp_{10}^4(2.18726)</math>
| <math>\exp_{10}^5(2.18726)</math>
| <math>\exp_{10}^6(2.18726)</math>
|- align=right
!scope="row"| 5
| 3,125 (5{{sup|5}})
| 1.91101 × 10{{sup|2,184}} (5{{sup|3,125}})
| <math>\exp_{10}^3(3.33928)</math> (10{{sup|1.33574×10{{sup|2,184}}}})
| <math>\exp_{10}^4(3.33928)</math>
| <math>\exp_{10}^5(3.33928)</math>
| <math>\exp_{10}^6(3.33928)</math>
|- align=right
!scope="row"| 6
| 46,656 (6{{sup|6}})
| 2.65912 × 10{{sup|36,305}} (6{{sup|46,656}})
| <math>\exp_{10}^3(4.55997)</math> (10{{sup|2.0692×10{{sup|36,305}}}})
| <math>\exp_{10}^4(4.55997)</math>
| <math>\exp_{10}^5(4.55997)</math>
| <math>\exp_{10}^6(4.55997)</math>
|- align=right
!scope="row"| 7
| 823,543 (7{{sup|7}})
| 3.75982 × 10{{sup|695,974}} (7<sup>823,543</sup>)
| <math>\exp_{10}^3(5.84259)</math> (3.17742 × 10{{sup|695,974}} digits)
| <math>\exp_{10}^4(5.84259)</math>
| <math>\exp_{10}^5(5.84259)</math>
| <math>\exp_{10}^6(5.84259)</math>
|- align=right
!scope="row"| 8
| 16,777,216 (8{{sup|8}})
| 6.01452 × 10{{sup|15,151,335}}
| <math>\exp_{10}^3(7.18045)</math> (5.43165 × 10{{sup|15,151,335}} digits)
| <math>\exp_{10}^4(7.18045)</math>
| <math>\exp_{10}^5(7.18045)</math>
| <math>\exp_{10}^6(7.18045)</math>
|- align=right
!scope="row"| 9
| 387,420,489 (9{{sup|9}})
| 4.28125 × 10{{sup|369,693,099}}
| <math>\exp_{10}^3(8.56784)</math> (4.08535 × 10{{sup|369,693,099}} digits)
| <math>\exp_{10}^4(8.56784)</math>
| <math>\exp_{10}^5(8.56784)</math>
| <math>\exp_{10}^6(8.56784)</math>
|- align=right
!scope="row"| 10
| 10,000,000,000 (10{{sup|10}})
| 10{{sup|10,000,000,000}}
| <math>\exp_{10}^4(1)</math> (10{{sup|10,000,000,000}} + 1 digits)
| <math>\exp_{10}^5(1)</math>
| <math>\exp_{10}^6(1)</math>
| <math>\exp_{10}^7(1)</math>
|}
'''Remark:''' If {{mvar|x}} does not differ from 10 by orders of magnitude, then for all <math>k \ge3,~ ^mx =\exp_{10}^k z,~z>1 ~\Rightarrow~^{m+1}x = \exp_{10}^{k+1} z' \text{ with }z' \approx z</math>. For example, <math>z - z' < 1.5\cdot 10^{-15} \text{ for } x = 3 = k,~ m = 4</math> in the above table, and the difference is even smaller for the following rows.

== Extensions ==
Tetration can be extended in two different ways; in the equation <math>^na\!</math>, both the base {{mvar|a}} and the height {{mvar|n}} can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the non-negative integers to different [[domain of a function|domains]], including <math>{^n 0}</math>, complex functions such as <math>{}^{n}i</math>, and heights of infinite {{mvar|n}}, the more limited properties of tetration reduce the ability to extend tetration.

=== Extension of domain for bases ===
==== Base zero ====
The exponential <math>0^0</math> is not consistently defined. Thus, the tetrations <math>\,{^{n}0}</math> are not clearly defined by the formula given earlier. However, <math>\lim_{x\rightarrow0} {}^{n}x</math> is well defined, and exists:<ref>{{cite web |url=https://math.blogoverflow.com/2015/01/05/climbing-the-ladder-of-hyper-operators-tetration/ |title=Climbing the ladder of hyper operators: tetration |series=Stack Exchange Mathematics Blog |website=math.blogoverflow.com |access-date=2019-07-25}}</ref>
:<math>\lim_{x\rightarrow0} {}^{n}x = \begin{cases}
  1, & n \text{ even} \\
  0, & n \text{ odd}
\end{cases}</math>

Thus we could consistently define <math>{}^{n}0 = \lim_{x\rightarrow 0} {}^{n}x</math>. This is analogous to defining <math>0^0 = 1</math>.

Under this extension, <math>{}^{0}0 = 1</math>, so the rule <math>{^{0}a} = 1</math> from the original definition still holds.

==== Complex bases ====
[[File:Tetration period.png|thumbnail|alt=A colorful graph that shows the period getting much larger|Tetration by period]]
[[File:Tetration escape.png|thumbnail|alt=A colorful graph that shows the escape getting much larger|Tetration by escape]]

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

: <math>i^{a+bi} = e^{\frac{1}{2}{\pi i} (a + bi)} = e^{-\frac{1}{2}{\pi b}} \left(\cos{\frac{\pi a}{2}} + i \sin{\frac{\pi a}{2}}\right)</math>

This suggests a recursive definition for {{math|{{sup|''n''+1}}''i'' {{=}} ''a′'' + ''b′i''}} given any {{math|{{sup|''n''}}''i'' {{=}} ''a'' + ''bi''}}:
: <math>\begin{align}
  a' &= e^{-\frac{1}{2}{\pi b}} \cos{\frac{\pi a}{2}} \\[2pt]
  b' &= e^{-\frac{1}{2}{\pi b}} \sin{\frac{\pi a}{2}}
\end{align}</math>

The following approximate values can be derived:
{| class="wikitable"
|+Values of tetration of complex bases
|-
!scope="col"| <math display="inline">{}^{n}i</math>
!scope="col"| Approximate value
|-
!scope="row"| <math display="inline">{}^{1}i = i</math>
| {{math|''i''}}
|-
!scope="row"| <math display="inline">{}^{2}i = i^{\left({}^{1}i\right)}</math>
| {{math|0.2079}}
|-
!scope="row"| <math display="inline">{}^{3}i = i^{\left({}^{2}i\right)}</math>
| {{math|0.9472 + 0.3208''i''}}
|-
!scope="row"| <math display="inline">{}^{4}i = i^{\left({}^{3}i\right)}</math>
| {{math|0.0501 + 0.6021''i''}}
|-
!scope="row"| <math display="inline">{}^{5}i = i^{\left({}^{4}i\right)}</math>
| {{math|0.3872 + 0.0305''i''}}
|-
!scope="row"| <math display="inline">{}^{6}i = i^{\left({}^{5}i\right)}</math>
| {{math|0.7823 + 0.5446''i''}}
|-
!scope="row"| <math display="inline">{}^{7}i = i^{\left({}^{6}i\right)}</math>
| {{math|0.1426 + 0.4005''i''}}
|-
!scope="row"| <math display="inline">{}^{8}i = i^{\left({}^{7}i\right)}</math>
| {{math|0.5198 + 0.1184''i''}}
|-
!scope="row"| <math display="inline">{}^{9}i = i^{\left({}^{8}i\right)}</math>
| {{math|0.5686 + 0.6051''i''}}
|}

Solving the inverse relation, as in the previous section, yields the expected {{math|{{sup|0}}''i'' {{=}} 1}} and {{math|{{sup|−1}}''i'' {{=}} 0}}, with negative values of {{mvar|n}} giving infinite results on the imaginary axis.{{citation needed|date=January 2025}} Plotted in the [[complex plane]], the entire sequence spirals to the limit {{math|0.4383 + 0.3606''i''}}, which could be interpreted as the value where {{mvar|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|date=July 2019}}

=== Extensions of the domain for different heights ===
==== Infinite heights ====
[[File:Infinite power tower.svg|thumb|alt=A line graph with a rapid curve upward as the base increases|<math>\textstyle \lim_{n\rightarrow \infty} {}^nx</math> of the infinitely iterated exponential converges for the bases <math>\textstyle \left(e^{-1}\right)^e \le x \le e^{\left(e^{-1}\right)}</math>]]
[[File:TetrationConvergence3D.png|thumbnail|alt=A three dimensional Cartesian graph with a point in the center|The function <math>\left| \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} \right|</math> on the complex plane, showing the real-valued infinitely iterated exponential function (black curve)]]

Tetration can be extended to [[Infinity|infinite]] heights; i.e., for certain {{mvar|a}} and {{mvar|n}} values in <math>{}^{n}a</math>, there exists a well defined result for an infinite {{mvar|n}}. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to [[infinity]]. For example, <math>\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}</math> 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:

: <math>\begin{align}
  \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.414}}}}}
    &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.63}}}} \\
    &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.76}}} \\
    &\approx \sqrt{2}^{\sqrt{2}^{1.84}} \\
    &\approx \sqrt{2}^{1.89} \\
    &\approx 1.93
\end{align}</math>

In general, the infinitely iterated exponential <math>x^{x^{\cdot^{\cdot^{\cdot}}}}\!\!</math>, defined as the limit of <math>{}^{n}x</math> as {{mvar|n}} goes to infinity, converges for {{math|''e''{{sup|−''e''}} ≤ ''x'' ≤ ''e''{{sup|1/''e''}}}}, roughly the interval from 0.066 to 1.44, a result shown by [[Leonhard Euler]].<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref> The limit, should it exist, is a positive real solution of the equation {{math|1=''y'' = ''x''{{sup|''y''}}}}. Thus, {{math|1=''x'' = ''y''{{sup|1/''y''}}}}. The limit defining the infinite exponential of {{mvar|x}} does not exist when {{math|''x'' > ''e''{{sup|1/''e''}}}} because the maximum of {{math|''y''{{sup|1/''y''}}}} is {{math|''e''{{sup|1/''e''}}}}. The limit also fails to exist when {{math|0 < ''x'' < ''e''{{sup|−''e''}}}}.

This may be extended to complex numbers {{mvar|z}} with the definition:
: <math>{}^{\infty}z = z^{z^{\cdot^{\cdot^{\cdot}}}} = \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} ~,</math>
where {{math|W}} represents [[Lambert's W function]].

As the limit {{math|1=''y'' = {{sup|∞}}''x''}} (if existent on the positive real line, i.e. for {{math|''e''{{sup|−''e''}} ≤ ''x'' ≤ ''e''{{sup|1/''e''}}}}) must satisfy {{math|1=''x''{{sup|''y''}} = ''y''}} we see that {{math|1=''x'' ↦ ''y'' = {{sup|∞}}''x''}} is (the lower branch of) the inverse function of {{math|1=''y'' ↦ ''x'' = ''y''{{sup|1/''y''}}}}.

==== Negative heights ====
We can use the recursive rule for tetration,
: <math>{^{k+1}a} = a^{\left({^{k}a}\right)},</math>

to prove <math>{}^{-1}a</math>:
: <math>^{k}a = \log_a \left(^{k+1}a\right);</math>

Substituting −1 for {{mvar|k}} gives
: <math>{}^{-1}a = \log_{a} \left({}^0 a\right) = \log_a 1 = 0</math>.<ref name="tetration extensions">{{cite web |url=http://www.mpmueller.net/reihenalgebra.pdf |last=Müller |first=M. |title=Reihenalgebra: What comes beyond exponentiation? |access-date=12 December 2018 |archive-date=2013-12-02  |archive-url=https://web.archive.org/web/20131202235053/http://www.mpmueller.net/reihenalgebra.pdf |url-status=dead }}</ref>

Smaller negative values cannot be well defined in this way. Substituting −2 for {{mvar|k}} in the same equation gives
: <math>{}^{-2}a = \log_{a} \left( {}^{-1}a \right) = \log_a 0 = -\infty</math>

which is not well defined. They can, however, sometimes be considered sets.<ref name="tetration extensions" />

For <math>n = 1</math>, any definition of <math>\,\! {^{-1}1}</math> is consistent with the rule because
: <math>{^{0}1} = 1 = 1^n</math> for any <math>\,\! n = {^{-1}1}</math>.

==== Linear approximation for real heights ====
[[File:Real-tetration.png|thumbnail|alt=A line graph with a figure drawn on it similar to an S-curve with values in the third quadrant going downward rapidly and values in the first quadrant going upward rapidly|<math>\,{}^{x}e</math> using linear approximation]]
A [[linear approximation]] (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
: <math>{}^{x}a \approx \begin{cases}
  \log_a\left(^{x+1}a\right) &  x \le -1 \\
                       1 + x & -1 < x \le 0 \\
    a^{\left(^{x-1}a\right)} &  0 < x
\end{cases}</math>

hence:
{| class="wikitable"
|+Linear approximation values
!scope="col"| Approximation
!scope="col"| Domain
|-
!scope="row"| <math display="inline">{}^x a \approx x + 1</math>
| for {{math|−1 < ''x'' < 0}}
|-
!scope="row"| <math display="inline">{}^x a \approx a^x</math>
| for {{math|0 < ''x'' < 1}}
|-
!scope="row"| <math display="inline">{}^x a \approx a^{a^{(x-1)}}</math>
| for {{math|1 < ''x'' < 2}}
|}

and so on. However, it is only piecewise differentiable; at integer values of {{mvar|x}}, the derivative is multiplied by <math>\ln{a}</math>. It is continuously differentiable for <math>x > -2</math> if and only if <math>a = e</math>. For example, using these methods <math>{}^\frac{\pi}{2}e \approx 5.868...</math> and <math>{}^{-4.3}0.5 \approx 4.03335...</math>

A main theorem in Hooshmand's paper<ref name="uxp" /> states: Let <math>0 < a \neq 1</math>. If <math>f:(-2, +\infty)\rightarrow \mathbb{R}</math> is continuous and satisfies the conditions:

* <math>f(x) = a^{f(x-1)} \;\; \text{for all} \;\; x > -1, \; f(0) = 1,</math>
* <math>f</math> is differentiable on {{open-open|−1, 0}},
* <math>f^\prime</math> is a nondecreasing or nonincreasing function on {{open-open|−1, 0}},
* <math>f^\prime \left(0^+\right) = (\ln a) f^\prime \left(0^-\right) \text{ or } f^\prime \left(-1^+\right) = f^\prime \left(0^-\right).</math>

then <math>f</math> is uniquely determined through the equation
: <math>f(x) = \exp^{[x]}_a \left(a^{(x)}\right) = \exp^{[x+1]}_a((x)) \quad \text{for all} \; \; x > -2,</math>

where <math>(x) = x - [x]</math> denotes the fractional part of {{mvar|x}} and <math>\exp^{[x]}_a</math> is the <math>[x]</math>-[[iterated function]] of the function <math>\exp_a</math>.

The proof is that the second through fourth conditions trivially imply that  {{mvar|f}}  is a linear function on {{closed-closed|−1, 0}}.

The linear approximation to natural tetration function <math>{}^xe</math> 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 <math> f: (-2, +\infty)\rightarrow \mathbb{R}</math> is a [[continuous function]] that satisfies:

* <math>f(x) = e^{f(x-1)} \;\; \text{for all} \;\; x > -1, \; f(0) = 1,</math>
* <math>f</math> is convex on {{open-open|−1, 0}},
* <math>f^\prime \left(0^-\right) \leq f^\prime \left(0^+\right).</math>

then <math>f = \text{uxp}</math>. [Here <math>f = \text{uxp}</math> 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 <math>f^\prime (-1^+) = f^\prime (0^+),</math> and then the convexity condition implies that <math>f</math> is linear on {{open-open|−1, 0}}.

Therefore, the linear approximation to natural tetration is the only solution of the equation <math>f(x) = e^{f(x-1)} \;\; (x > -1)</math> and <math>f(0) = 1</math> which is [[convex function|convex]] on {{open-open|−1, +∞}}. All other sufficiently-differentiable solutions must have an [[inflection point]] on the interval {{open-open|−1, 0}}.

==== Higher order approximations for real heights ====
[[File:Approximations of 0.5 tetratrated to the x.png|thumb|alt=A pair of line graphs, with one drawn in blue looking similar to a sine wave that has a decreasing amplitude as the values along the x-axis increase and the second is a red line that directly connects points along these curves with line segments|A comparison of the linear and quadratic approximations (in red and blue respectively) of the function <math>^{x}0.5</math>, from {{math|1=''x'' = −2}} to {{math|1=''x'' = 2}}]]
Beyond linear approximations, a [[quadratic approximation]] (to the differentiability requirement) is given by:
: <math>{}^{x}a \approx \begin{cases}
  \log_a\left({}^{x+1}a\right) & x \le -1 \\
  1 + \frac{2\ln(a)}{1 \;+\; \ln(a)}x - \frac{1 \;-\; \ln(a)}{1 \;+\; \ln(a)}x^2 & -1 < x \le 0 \\
  a^{\left({}^{x-1}a\right)} & x >0
\end{cases}</math>

which is differentiable for all <math>x > 0</math>, but not twice differentiable. For example, <math>{}^\frac{1}{2}2 \approx 1.45933...</math> If <math>a = e</math> this is the same as the linear approximation.<ref name="uwu" />

Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where <math>\left(a^\frac{1}{n}\right)^n = a</math>. Namely,
: <math>
  {}^n\left({}^\frac{1}{n} a\right)
  = \underbrace{
      \left({}^\frac{1}{n}a\right)^{
        \left({}^\frac{1}{n}a\right)^{
          \cdot^{\cdot^{\cdot^{\cdot^{
            \left({}^\frac{1}{n}a\right)
          }}}}
        }
      }
    }_n
  \neq a
</math>.

Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree {{mvar|n}} also exist, although they are much more unwieldy.<ref name="uwu" /><ref name=SolveAnalyt>Andrew Robbins. [https://web.archive.org/web/20090201164821/http://tetration.itgo.com/paper.html Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm]. The extensions are found in part two of the paper, "Beginning of Results".</ref>

=== Complex heights ===
[[File:Tetration analytic extension.svg|thumb|alt=A complex graph showing mushrooming values along the x-axis|Drawing of the analytic extension <math>f = F(x+{\rm i}y)</math> of tetration to the complex plane. Levels <math>|f| = 1, e^{\pm 1}, e^{\pm 2}, \ldots</math> and levels <math>\arg(f) = 0, \pm 1, \pm 2, \ldots</math> are shown with thick curves.]]

In 2017, it was proved<ref name="PAU10">{{cite journal
  |author-first1=W.
  |author-last1=Paulsen
  |author-first2=S.
  |author-last2=Cowgill
  |title=Solving <math>F(z+1) = b^{F(z)}</math> in the complex plane
  |journal=Advances in Computational Mathematics
  |volume=43
  |pages=1–22
  |date=March 2017
  |doi=10.1007/s10444-017-9524-1
  |url=http://myweb.astate.edu/wpaulsen/tetration2.pdf
  |s2cid=9402035
}}</ref> that there exists a unique function <math>F</math> satisfying
<math>F(z + 1) = \exp\bigl(F(z)\bigr)</math> (equivalently <math>F(z+1) = b^{F(z)}</math> when <math>b=e</math>), with the auxiliary conditions
<math>F(0) = 1</math>, and
<math>F(z) \to \xi_{\pm}</math> (the attracting/repelling fixed points of the logarithm, roughly <math>0.318 \pm 1.337\,\mathrm{i}</math>) as <math>z \to \pm i\infty</math>.  Moreover, <math>F</math> is holomorphic on all of <math>\mathbb{C}</math> except for the cut along the real axis at <math>z \le -2</math>.  This construction was first conjectured by Kouznetsov (2009)<ref name="MOC09">{{cite journal
  |author-first=D.
  |author-last=Kouznetsov
  |title=Solution of <math>F(z + 1) = \exp(F(z))</math> in complex <math>z</math>-plane
  |journal=Mathematics of Computation
  |volume=78
  |issue=267
  |pages=1647–1670
  |date=July 2009
  |doi=10.1090/S0025-5718-09-02188-7
  |url=http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/S0025-5718-09-02188-7.pdf
}}</ref> and rigorously carried out by Kneser in 1950.<ref name="hellmuth50">{{cite journal
  |author-first=H.
  |author-last=Kneser
  |title=Reelle analytische Lösungen der Gleichung <math>\varphi(\varphi(x)) = e^x</math> und verwandter Funktionalgleichungen
  |journal=Journal für die reine und angewandte Mathematik
  |volume=187
  |pages=56–67
  |date=1950
  |language=de
}}</ref>  Paulsen &amp; Cowgill’s proof extends Kneser’s original construction to any base <math>b>e^{1/e}\approx1.445</math>, and subsequent work showed how to allow <math>b \in \mathbb{C}</math> with <math>|b|>e^{1/e}</math>.<ref name="PAU18">{{cite journal
  |author-first=W.
  |author-last=Paulsen
  |title=Tetration for complex bases
  |journal=Advances in Computational Mathematics
  |volume=45
  |pages=243–267
  |date=June 2018
  |doi=10.1007/s10444-018-9615-7
}}</ref>

In May 2025, Vey gave a unified, holomorphic extension for arbitrary complex bases <math>b\in \mathbb{C}\setminus\{0,1\}</math> and complex heights <math>z\in\mathbb{C}</math> by means of Schröder’s equation.  In particular, one constructs a linearizing coordinate near the attracting (or repelling) fixed point of the map <math>f(w)=b^w</math>, and then patches together two analytic expansions (one around each fixed point) to produce a single function <math>F_{b}(z)</math> that satisfies
<math>F_{b}(z+1)=b^{\,F_{b}(z)}</math> and <math>F_{b}(0)=1</math> on all of <math>\mathbb{C}</math>.  The key step is to define
<math>\displaystyle
  \Phi_{b}(w)=\lim_{n\to\infty}\;s^{n}\Bigl(f^{\circ n}(w)-\alpha\Bigr),
</math>
where <math>\alpha</math> is a fixed point of <math>f(w)=b^w</math>, <math>s = f'(\alpha)</math>, and <math>f^{\circ n}</math> denotes <math>n</math>-fold iteration.  One then solves Schröder’s functional equation
<math>\Phi_{b}\bigl(b^{\,w}\bigr)\;=\;s\;\Phi_{b}(w)</math>
locally (for <math>w</math> near <math>\alpha</math>), extends both branches holomorphically, and glues them so that there is no monodromy except the known cut-lines.  Vey also provides explicit series for the coefficients <math>a_{n}^{(b)}</math> in the local Schröder expansion:
<math>\Phi_{b}(w)
  = \sum_{n=0}^{\infty} a_{n}^{(b)}\,(w-\alpha)^{n},
</math>
and gives rigorous bounds proving factorial convergence of <math>a_{n}^{(b)}</math>.<ref name="VEY25">{{cite web 
  |author-first=Vincent
  |author-last=Vey
  |title=Holomorphic Extension of Tetration to Complex Bases and Heights via Schröder’s Equation
  |date=May 2025
  |url=http://dx.doi.org/10.13140/RG.2.2.10348.48008
}}</ref>

Using Kneser’s (and Vey’s) tetration, example values include
<math>{}^{\tfrac{\pi}{2}}e \approx 5.82366\ldots</math>,
<math>{}^{\tfrac{1}{2}}2 \approx 1.45878\ldots</math>, and
<math>{}^{\tfrac{1}{2}}e \approx 1.64635\ldots</math>.

The requirement that tetration be holomorphic on all of <math>\mathbb{C}</math> (except for the known cuts) is essential for uniqueness.  If one relaxes holomorphicity, there are infinitely many real‐analytic “solutions” obtained by pre‐ or post‐composing with almost‐periodic perturbations.  For example, for any fast‐decaying real sequences <math>\{\alpha_{n}\}</math> and <math>\{\beta_{n}\}</math>, one can set
<math>
  S(z)
  =
  F_{b}\Bigl(\,
    z
    +\sum_{n=1}^{\infty}\sin(2\pi n\,z)\,\alpha_{n}
    +\sum_{n=1}^{\infty}\bigl[1 - \cos(2\pi n\,z)\bigr]\,\beta_{n}
  \Bigr),
</math>
which still satisfies <math>S(z+1)=b^{S(z)}</math> and <math>S(0)=1</math>, but has additional singularities creeping in from the imaginary direction.

<syntaxhighlight lang="wikitext">
<!-- Example of “calling” Vey’s solution in pseudocode (series form) -->
function ComplexTetration(b, z):
    # 1) Find attracting fixed point alpha of w ↦ b^w
    α ← the unique solution of α = b^α near the real line
    # 2) Compute multiplier s = b^α · ln(b)
    s ← b**α * log(b)
    # 3) Solve Schröder’s equation coefficients {a_n} around α:
    #    Φ_b(w) = ∑_{n=0}^∞ a_n · (w − α)^n,   Φ_b(b^w) = s · Φ_b(w)
    {a_n} ← SolveLinearSystemSchroeder(b, α, s)
    # 4) Define inverse φ_b⁻¹ via the local power series around 0
    φ_inv(u) = α + ∑_{n=1}^∞ c_n · u^n   # (coefficients c_n from series inversion)
    # 5) Put F_b(z) = φ_b⁻¹(s^(-z) · Φ_b(1))
    return φ_inv( s^(−z) * ∑_{n=0}^∞ a_n · (1 − α)^n )
</syntaxhighlight>
<!-- End of code example -->
<!-- Note: Vey’s actual paper shows how to bound |a_n| ≲ C·n!·R⁻ⁿ so that the
     series converges for all w in a suitable neighborhood of α; see Vey 2025. -->
=== Ordinal tetration ===
Tetration can be defined for [[ordinal numbers]] via [[transfinite induction]].  For all {{math|''&alpha;''}} and all {{math|''&beta;'' > 0}}:
<math display=block>{}^0\alpha = 1</math>
<math display=block>{}^\beta\alpha = \sup(\{\alpha^{{}^\gamma\alpha} : \gamma < \beta\})\,.</math>

== Non-elementary recursiveness ==
Tetration (restricted to <math>\mathbb{N}^2</math>) is not an [[ELEMENTARY|elementary recursive function]]. One can prove by induction that for every elementary recursive function {{mvar|f}}, there is a constant {{mvar|c}} such that
: <math>f(x) \leq \underbrace{2^{2^{\cdot^{\cdot^{x}}}}}_c.</math>
We denote the right hand side by <math>g(c, x)</math>. Suppose on the contrary that tetration is elementary recursive. <math>g(x, x)+1</math> is also elementary recursive. By the above inequality, there is a constant {{mvar|c}} such that <math>g(x,x) +1 \leq g(c, x)</math>. By letting <math>x=c</math>, we have that <math>g(c,c) + 1 \leq g(c, c)</math>, a contradiction.
<!-- who originally posted this proof? -->

== Inverse operations ==
[[Exponentiation]] has two inverse operations; [[Nth root|roots]] and [[logarithm]]s. Analogously, the [[Inverse function|inverses]] of tetration are often called the '''''super-root''''', and the '''''super-logarithm''''' (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function <math>{^3}y=x</math>, the two inverses are the cube super-root of {{mvar|y}} and the super-logarithm base&nbsp;{{mvar|y}} of {{mvar|x}}.

=== Super-root ===
{{Redirect|Super-root|the directory supported by some Unixes|super-root (computing)}}
The super-root is the inverse operation of tetration with respect to the base: if <math>^n y = x</math>, then {{mvar|y}} is an {{mvar|n}}th super-root of {{mvar|x}} (<math>\sqrt[n]{x}_s</math> or <math>\sqrt[4]{x}_s</math>).

For example,
: <math>^4 2 = 2^{2^{2^{2}}} = 65{,}536</math>

so 2 is the 4th super-root of 65,536 <math>\left(\sqrt[4]{65{,}536}_s =2\right)</math>.

==== Square super-root ====
[[File:The graph y = √x(s).svg|thumb|alt=A curve that starts at (0,1), bends slightly to the right and then bends back dramatically to the left as the values along the x-axis increase|The graph <math>y = \sqrt{x}_s</math>]]
The ''2nd-order super-root'', ''square super-root'', or ''super square root'' has two equivalent notations, <math>\mathrm{ssrt}(x)</math> and <math>\sqrt{x}_s</math>. It is the inverse of <math>^2 x = x^x</math> and can be represented with the [[Lambert W function]]:<ref name="Corless">{{cite journal |last1=Corless |first1=R. M. |last2=Gonnet |first2=G. H. |last3=Hare |first3=D. E. G. |last4=Jeffrey |first4=D. J. |last5=Knuth |first5=D. E. |author5-link=Donald Knuth |title=On the Lambert W function |journal=Advances in Computational Mathematics |volume=5 |page=333 |date=1996 |url=http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps<!-- or http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf --> |format=[[PostScript]]<!-- or PDF --> |doi=10.1007/BF02124750 |arxiv=1809.07369 |s2cid=29028411}}</ref>

: <math>\mathrm{ssrt}(x)=\exp(W(\ln x))=\frac{\ln x}{W(\ln x)}</math> or
: <math>\sqrt{x}_s=e^{W(\ln x)}</math>

The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when <math>y = \mathrm{ssrt}(x)</math>:

: <math>\sqrt[y]{x} = \log_y x</math>

Like [[square root]]s, the square super-root of {{mvar|x}} may not have a single solution. Unlike square roots, determining the number of square super-roots of {{mvar|x}} may be difficult. In general, if <math>e^{-1/e}<x<1</math>, then {{mvar|x}} has two positive square super-roots between 0 and 1 calculated using formulas:<math>\sqrt{x}_s=\left\{e^{W_{-1}(\ln x)};e^{W_{0}(\ln x)}\right\}</math>; and if <math>x > 1</math>, then {{mvar|x}} has one positive square super-root greater than 1 calculated using formulas: <math>\sqrt{x}_s=e^{W_{0}(\ln x)}</math>. If {{mvar|x}} is positive and less than <math>e^{-1/e}</math> it does not have any [[real number|real]] square super-roots, but the formula given above yields countably infinitely many [[complex number|complex]] ones for any finite {{mvar|x}} not equal to 1.<ref name="Corless" /> The function has been used to determine the size of [[data cluster]]s.<ref>Krishnam, R. (2004), "[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.8594 Efficient Self-Organization Of Large Wireless Sensor Networks]" – Dissertation, BOSTON UNIVERSITY, COLLEGE OF ENGINEERING. pp. 37–40</ref>

At <math> x = 1 </math>:
{{center|<math display="block"> \mathrm{ssrt}(x) = 1 + (x-1) -(x-1)^2 + \frac{3}{2} (x-1)^3 - \frac{17}{6} (x-1)^4 + \frac{37}{6}(x-1)^5 - \frac{1759}{120}(x-1)^6 + \frac{13279}{360} (x-1)^7 + \mathcal O {\left((x-1)^8 \right)} </math>}}


==== Other super-roots ====
[[File:Cube super root.png|thumb|alt=A line graph that starts at the origin and quickly makes an asymptote toward 2 as the value along the x-axis increases|The graph <math>y=\sqrt[3]{x}_s</math>]]
One of the simpler and faster formulas for a third-degree super-root is the recursive formula. If <math>y = x^{x^x}</math> then one can use:
* <math>x_0 = 1</math>
* <math>x_{n+1} = \exp(W(W(x_n\ln y)))</math>

For each integer {{math|''n'' > 2}}, the function {{math|''{{sup|n}}x''}} is defined and increasing for {{math|''x'' ≥ 1}}, and {{math|1={{sup|''n''}}1 = 1}}, so that the {{mvar|n}}th super-root of {{mvar|x}}, <math>\sqrt[n]{x}_s</math>, exists for {{math|''x'' ≥ 1}}.

However, if the [[#Linear approximation for real heights|linear approximation above]] is used, then <math> ^y x = y + 1</math> if {{math|−1 < ''y'' ≤ 0}}, so <math> ^y \sqrt{y + 1}_s </math> cannot exist.

In the same way as the square super-root, terminology for other super-roots can be based on the [[nth root|normal roots]]: "cube super-roots" can be expressed as <math>\sqrt[3]{x}_s</math>; the "4th super-root" can be expressed as <math>\sqrt[4]{x}_s</math>; and the "{{mvar|n}}th super-root" is <math>\sqrt[n]{x}_s</math>. Note that <math>\sqrt[n]{x}_s</math> may not be uniquely defined, because there may be more than one {{mvar|n}}{{sup|th}} root. For example, {{mvar|x}} has a single (real) super-root if {{mvar|n}} is ''odd'', and up to two if {{mvar|n}} is ''even''.{{Citation needed|date=October 2009}}

Just as with the extension of tetration to infinite heights, the super-root can be extended to {{math|1=''n'' = ∞}}, being well-defined if {{math|1/''e'' ≤ ''x'' ≤ ''e''}}. Note that <math> x = {^\infty y} = y^{\left[^\infty y\right]} = y^x,</math> and thus that <math> y = x^{1/x} </math>. Therefore, when it is well defined, <math> \sqrt[\infty]{x}_s = x^{1/x} </math> and, unlike normal tetration, is an [[elementary function]]. For example, <math>\sqrt[\infty]{2}_s = 2^{1/2} = \sqrt{2}</math>.

It follows from the [[Gelfond–Schneider theorem]] that super-root <math>\sqrt{n}_s</math> for any positive integer {{mvar|n}} is either integer or [[Transcendental number|transcendental]], and <math>\sqrt[3]{n}_s</math> is either integer or irrational.<ref name="condor.depaul.edu" /> It is still an open question whether irrational super-roots are transcendental in the latter case.

=== Super-logarithm ===
{{Main|Super-logarithm}}
Once a continuous increasing (in {{mvar|x}}) definition of tetration, {{math|{{sup|''x''}}''a''}}, is selected, the corresponding super-logarithm <math>\operatorname{slog}_ax</math> or <math>\log^4_ax</math> is defined for all real numbers {{mvar|x}}, and {{math|''a'' > 1}}.

The function {{math|slog<sub>''a''</sub> ''x''}} satisfies:
: <math>\begin{align}
\operatorname{slog}_a {^x a} &= x \\
\operatorname{slog}_a a^x &= 1 + \operatorname{slog}_a x \\
\operatorname{slog}_a x &= 1 + \operatorname{slog}_a \log_a x \\
\operatorname{slog}_a x &\geq -2
\end{align}
</math>

== Open questions ==
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 [[integer]]s and [[irrational number]]s:
* It is not known whether there is an integer <math> n \ge 4</math> for which {{math|{{sup|''n''}}[[pi|&pi;]]}} is an integer, because we could not calculate precisely enough the numbers of digits after the decimal points of <math>\pi</math>.<ref>{{cite web |last1=Bischoff |first1=Manon |title=A Wild Claim about the Powers of Pi Creates a Transcendental Mystery |url=https://www.scientificamerican.com/article/a-wild-claim-about-the-powers-of-pi-creates-a-transcendental-mystery/ |website=Scientific American |access-date=23 April 2024 |archive-url=https://web.archive.org/web/20240424025038/https://www.scientificamerican.com/article/a-wild-claim-about-the-powers-of-pi-creates-a-transcendental-mystery/ |archive-date=24 April 2024 |language=en |date=24 January 2024 |url-status=live}}</ref>{{Additional citation needed|date=September 2024|reason=Citation needed for the e and general pi^^n claims.}} It is similar for {{math|{{sup|''n''}}[[e (mathematical constant)|''e'']]}} for <math> n \ge 5</math>, as we are not aware of any other methods besides some direct computation. In fact, since <math> \log_{10}(e) \cdot {}^{3}e = 1656520.36764</math>, then <math>{}^{4}e > 2\cdot 10^{1656520}</math>. Given <math>{}^{3}\pi < 1.35\cdot 10^{18} \ll 10^{1656520} </math> and <math>\pi < e^2</math>, then <math>{}^{4}\pi < {}^{n}e</math> for <math> n \ge 5</math>. It is believed that {{math|{{sup|''n''}}[[e (mathematical constant)|''e'']]}} is not an integer for any positive integer {{mvar|n}}, due to the [[algebraic independence]] of <math>e, {}^{2}e, {}^{3}e, \dots</math>, given [[Schanuel's conjecture]].<ref name="Cheng">{{cite journal |last1=Cheng |first1=Chuangxun |last2=Dietel |first2=Brian |last3=Herblot |first3=Mathilde |last4=Huang |first4=Jingjing |last5=Krieger |first5=Holly |last6=Marques |first6=Diego |last7=Mason |first7=Jonathan |last8=Mereb |first8=Martin |last9=Wilson |first9=S. Robert |date=2009 |title=Some consequences of Schanuel's conjecture |journal=Journal of Number Theory |volume=129 |issue=6 |pages=1464–1467 |doi=10.1016/j.jnt.2008.10.018 |arxiv=0804.3550}}</ref>
* It is not known whether {{math|{{sup|''n''}}''q''}} is rational for any positive integer {{mvar|n}} and positive non-integer rational {{mvar|q}}.<ref name="condor.depaul.edu">[http://condor.depaul.edu/mash/atotheamg.pdf Marshall, Ash J., and Tan, Yiren, "A rational number of the form {{math|''a''{{sup|''a''}}}} with {{mvar|a}} irrational", Mathematical Gazette 96, March 2012, pp. 106–109.]</ref> For example, it is not known whether the positive root of the equation {{math|1={{sup|4}}''x'' = 2}} is a rational number.{{citation needed|date=October 2020}}
* It is not known whether {{math|{{sup|e}}&pi;}} or {{math|{{sup|π}}e}} (defined using Kneser's extension) are rationals or not.

== Applications ==

For each graph ''H'' on ''h'' vertices and each {{math|ε &gt; 0}}, define
:<math>D=2\uparrow\uparrow5h^4\log(1/\varepsilon).</math>
Then each graph ''G'' on ''n'' vertices with at most {{math|n{{sup|h}}/D}} copies of ''H'' can be made ''H''-free by removing at most {{math|εn{{sup|2}}}} edges.<ref>Jacob Fox, [https://arxiv.org/abs/1006.1300 A new proof of the graph removal lemma], arXiv preprint (2010). arXiv:1006.1300 [math.CO]</ref>

==See also==
{{Commons category|Tetration|lcfirst=yes}}
{{Wiktionary}}

*[[Ackermann function]]
*[[Big O notation]]
*[[Double exponential function]]
*[[Hyperoperation]]
*[[Iterated logarithm]]
*[[Symmetric level-index arithmetic]]

== Notes ==
{{Reflist|group="nb"|refs=
<!-- <ref group="nb" name="NB_Rucker">[[Rudolf von Bitter Rucker]]'s (1982) notation {{math|{{i sup|''n''}}''x''}}, as introduced by Hans Maurer (1901) and [[Reuben Louis Goodstein]] (1947) for tetration, must not be confused with [[Alfred Pringsheim]]'s and [[Jules Molk]]'s (1907) notation {{math|{{i sup|''n''}}''f''(''x'')}} to denote iterated [[function composition]]s, nor with [[David Patterson Ellerman]]'s (1995) {{math|{{i sup|''n''}}''x''}} pre-superscript notation for [[nth root|roots]].See {{cite book |title=Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics |chapter=Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics: Series Chauvinsism |series=G – Reference, Information and Interdisciplinary Subjects Series |work=The worldly philosophy: studies in intersection of philosophy and economics |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |edition=illustrated |publisher=[[Rowman & Littlefield Publishers, Inc.]] |date=1995-03-21 |isbn=0-8476-7932-2 |pages=237–268 [239] |url=http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |chapter-url=https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237 |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20160305012729/http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |archive-date=2016-03-05}} [https://web.archive.org/web/20150917191423/http://www.ellerman.org/Davids-Stuff/Maths/sp_math.doc] (271 pages){{cite web |title=Introduction to Series-Parallel Duality |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |publisher=[[University of California at Riverside]] |date=May 2004 |orig-year=1995-03-21 |citeseerx=10.1.1.90.3666 |url=http://www.ellerman.org/wp-content/uploads/2012/12/Series-Parallel-Duality.CV_.pdf |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20190810011716/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf |archive-date=2019-08-10}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf] (24 pages)</ref> -->
}}

== References ==
{{Reflist}}

== External Links ==
* Daniel Geisler, ''[http://www.tetration.org/ Tetration]''
* Ioannis Galidakis, ''[https://ingalidakis.com/math/exponents4.html On extending hyper4 to nonintegers]'' (undated, 2006 or earlier) ''(A simpler, easier to read review of the next reference)''
* Ioannis Galidakis, ''[https://web.archive.org/web/20060525195301/http://ioannis.virtualcomposer2000.com/math/papers/Extensions.pdf On Extending hyper4 and Knuth's Up-arrow Notation to the Reals]'' (undated, 2006 or earlier).
* Robert Munafo, ''[http://mrob.com/pub/math/hyper4.html#real-hyper4 Extension of the hyper4 function to reals]'' ''(An informal discussion about extending tetration to the real numbers.)''
* Lode Vandevenne, ''[http://groups.google.com/group/sci.math/browse_frm/thread/39a7019f9051c5d7/8c1c4facb7e4bd6d#8c1c4facb7e4bd6d Tetration of the Square Root of Two]''. (2004). ''(Attempt to extend tetration to real numbers.)''
* Ioannis Galidakis, ''[https://ingalidakis.com/math/index.html Mathematics]'', ''(Definitive list of references to tetration research. Much information on the Lambert W function, Riemann surfaces, and analytic continuation.)''
* Joseph MacDonell, ''[http://www.faculty.fairfield.edu/jmac/ther/tower.htm Some Critical Points of the Hyperpower Function] {{Webarchive|url=https://web.archive.org/web/20100117161604/http://www.faculty.fairfield.edu/jmac/ther/tower.htm |date=2010-01-17  }}''.
* Dave L. Renfro, ''[http://mathforum.org/discuss/sci.math/t/350321 Web pages for infinitely iterated exponentials]''
* {{cite journal |author-last=Knobel |author-first=R. |date=1981 |title=Exponentials Reiterated |journal=[[American Mathematical Monthly]] |volume=88 |issue=4 |pages=235–252 |doi=10.1080/00029890.1981.11995239}}
* Hans Maurer, "Über die Funktion <math>y=x^{[x^{[x(\cdots)]}]}</math> für ganzzahliges Argument (Abundanzen)." ''Mittheilungen der Mathematische Gesellschaft in Hamburg'' '''4''', (1901), p.&nbsp;33–50. ''(Reference to usage of <math>\ {^{n} a}</math> from Knobel's paper.)''
* ''[http://ariwatch.com/VS/Algorithms/TheFourthOperation.htm The Fourth Operation]''
* Luca Moroni, [https://arxiv.org/abs/1908.05559 ''The strange properties of the infinite power tower''] (https://arxiv.org/abs/1908.05559)
* Vincent Vey, ''[Preprint (Mai 2025): Holomorphic Extension of Tetration to Complex Bases and Heights via Schröder’s Equation]'' [http://dx.doi.org/10.13140/RG.2.2.10348.48008 DOI]
== Further reading ==
* {{MathWorld |id=PowerTower |title=Power Tower |author=Galidakis, Ioannis<!-- |author-last1=Galidakis |author-first1=Ioannis --> |author-last2=Weisstein |author-first2=Eric Wolfgang |author-link2=Eric Wolfgang Weisstein |access-date=5 July 2019}}

{{Hyperoperations}}
{{Large numbers}}

[[Category:Exponentials]]
[[Category:Operations on numbers]]
[[Category:Large numbers]]