Tetration#Extension to real heights

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as $a\text{'}\; =\; a\; +\; 1$, is considered to be the zeroth operation.

1. Addition $$a\; +\; n\; =\; a\; +\; \backslash underbrace\{1\; +\; 1\; +\; \backslash cdots\; +\; 1\}\_n$$ {{mvar|n}} copies of 1 added to {{mvar|a}} combined by succession.
2. Multiplication $$a\; \backslash times\; n\; =\; \backslash underbrace\{a\; +\; a\; +\; \backslash cdots\; +\; a\}\_n$$ {{mvar|n}} copies of {{mvar|a}} combined by addition.
3. Exponentiation $$a^n\; =\; \backslash underbrace\{a\; \backslash times\; a\; \backslash times\; \backslash cdots\; \backslash times\; a\}\_n$$ {{mvar|n}} copies of {{mvar|a}} combined by multiplication.
4. Tetration $$\{^\{n\}a\}\; =\; \backslash underbrace\{a^\{a^\{\backslash cdot^\{\backslash cdot^\{a\}\}\}\}\}\_n$$ {{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, $a\_\{n+1\}\; =\; a\_n\; +\; 1$, is the most basic operation; while addition ($a\; +\; n$) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of $n$ successors of $a$; multiplication ($a\; \backslash times\; n$) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving $n$ numbers of $a$. Exponentiation can be thought of as a chained multiplication involving $n$ numbers of $a$ and tetration ($^\{n\}a$) as a chained power involving $n$ numbers $a$. Each of the operations above are defined by iterating the previous one;Neyrinck, Mark. [http://skysrv.pha.jhu.edu/~neyrinck/extessay.pdf An Investigation of Arithmetic Operations.] Retrieved 9 January 2019. however, unlike the operations before it, tetration is not an elementary function.

The parameter $a$ is referred to as the base, while the parameter $n$ 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 $a\; >\; 0$ and non-negative integer $n\; \backslash ge\; 0$, we can define $\backslash ,\backslash !\; \{^\{n\}a\}$ recursively as:

:

The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as $^\{0\}a$, $^\{-1\}a$, and $^\{i\}a$ as well – many of these extensions are areas of active research.

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{{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 }} (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.{{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 }} It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
• The term hyperpower{{cite journal |author=J. F. MacDonnell |title=Somecritical points of the hyperpower function $x^\{x^\{\backslash dots\}\}$ |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 }} 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{{MathWorld |urlname=PowerTower |title=Power Tower}} is occasionally used, in the form "the power tower of order {{mvar|n}}" for $\{\backslash \; \backslash atop\; \{\backslash \; \}\}\; \{\{\backslash underbrace\{a^\{a^\{\backslash cdot^\{\backslash cdot^\{a\}\}\}\}\}\}\; \backslash atop\; n\}$. Exponentiation is easily misconstrued: note that the operation of raising to a power is right-associative (see below). Tetration is iterated exponentiation (call this 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 notational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:

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 powers or iterated exponentials.

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

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

: $\backslash exp\_a^n(x)\; =\; a^\{a^\{\backslash cdot^\{\backslash cdot^\{a^x\}\}\}\}$ with {{mvar|n}} {{mvar|a}}s.

There are not as many notations for iterated exponentials, but here are a few:

Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate. 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.

Remark: If {{mvar|x}} does not differ from 10 by orders of magnitude, then for all $k\; \backslash ge3,~\; ^mx\; =\backslash exp\_\{10\}^k\; z,~z>1\; ~\backslash Rightarrow~^\{m+1\}x\; =\; \backslash exp\_\{10\}^\{k+1\}\; z\text{'}\; \backslash text\{\; with\; \}z\text{'}\; \backslash approx\; z$. For example, in the above table, and the difference is even smaller for the following rows.

Tetration can be extended in two different ways; in the equation $^na\backslash !$, 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 domains, including $\{^n\; 0\}$, complex functions such as $\{\}^\{n\}i$, and heights of infinite {{mvar|n}}, the more limited properties of tetration reduce the ability to extend tetration.

The exponential $0^0$ is not consistently defined. Thus, the tetrations $\backslash ,\{^\{n\}0\}$ are not clearly defined by the formula given earlier. However, $\backslash lim\_\{x\backslash rightarrow0\}\; \{\}^\{n\}x$ is well defined, and exists:{{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}}

:$\backslash lim\_\{x\backslash rightarrow0\}\; \{\}^\{n\}x\; =\; \backslash begin\{cases\}$

1, & n \text{ even} \\

0, & n \text{ odd}

\end{cases}

Thus we could consistently define $\{\}^\{n\}0\; =\; \backslash lim\_\{x\backslash rightarrow\; 0\}\; \{\}^\{n\}x$. This is analogous to defining $0^0\; =\; 1$.

Under this extension, $\{\}^\{0\}0\; =\; 1$, so the rule $\{^\{0\}a\}\; =\; 1$ from the original definition still holds.

File:Tetration period.png

File:Tetration escape.png

Since complex numbers 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:

: $i^\{a+bi\}\; =\; e^\{\backslash frac\{1\}\{2\}\{\backslash pi\; i\}\; (a\; +\; bi)\}\; =\; e^\{-\backslash frac\{1\}\{2\}\{\backslash pi\; b\}\}\; \backslash left(\backslash cos\{\backslash frac\{\backslash pi\; a\}\{2\}\}\; +\; i\; \backslash sin\{\backslash frac\{\backslash pi\; a\}\{2\}\}\backslash right)$

This suggests a recursive definition for {{math|{{sup|n+1}}i {{=}} a′ + b′i}} given any {{math|{{sup|n}}i {{=}} a + bi}}:

: $\backslash begin\{align\}$

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}

The following approximate values can be derived:

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 + 0.3606i}}, 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}}

File:Infinite power tower.svg

File:TetrationConvergence3D.png

Tetration can be extended to infinite heights; i.e., for certain {{mvar|a}} and {{mvar|n}} values in $\{\}^\{n\}a$, 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, $\backslash sqrt\{2\}^\{\backslash sqrt\{2\}^\{\backslash sqrt\{2\}^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot\}\}\}\}\}$ 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:

: $\backslash begin\{align\}$

\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}

In general, the infinitely iterated exponential $x^\{x^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot\}\}\}\}\backslash !\backslash !$, defined as the limit of $\{\}^\{n\}x$ 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.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]) 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:

: $\{\}^\{\backslash infty\}z\; =\; z^\{z^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot\}\}\}\}\; =\; \backslash frac\{\backslash mathrm\{W\}(-\backslash ln\{z\})\}\{-\backslash ln\{z\}\}\; ~,$

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}}}}.

We can use the recursive rule for tetration,

: $\{^\{k+1\}a\}\; =\; a^\{\backslash left(\{^\{k\}a\}\backslash right)\},$

to prove $\{\}^\{-1\}a$:

: $^\{k\}a\; =\; \backslash log\_a\; \backslash left(^\{k+1\}a\backslash right);$

Substituting −1 for {{mvar|k}} gives

: $\{\}^\{-1\}a\; =\; \backslash log\_\{a\}\; \backslash left(\{\}^0\; a\backslash right)\; =\; \backslash log\_a\; 1\; =\; 0$.{{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 }}

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

: $\{\}^\{-2\}a\; =\; \backslash log\_\{a\}\; \backslash left(\; \{\}^\{-1\}a\; \backslash right)\; =\; \backslash log\_a\; 0\; =\; -\backslash infty$

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

For $n\; =\; 1$, any definition of $\backslash ,\backslash !\; \{^\{-1\}1\}$ is consistent with the rule because

: $\{^\{0\}1\}\; =\; 1\; =\; 1^n$ for any $\backslash ,\backslash !\; n\; =\; \{^\{-1\}1\}$.

File:Real-tetration.png

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

: $\{\}^\{x\}a\; \backslash approx\; \backslash begin\{cases\}$

\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}

hence:

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

A main theorem in Hooshmand's paper states: Let . If $f:(-2,\; +\backslash infty)\backslash rightarrow\; \backslash mathbb\{R\}$ is continuous and satisfies the conditions:

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

then $f$ is uniquely determined through the equation

: $f(x)\; =\; \backslash exp^\{[x]\}\_a\; \backslash left(a^\{(x)\}\backslash right)\; =\; \backslash exp^\{[x+1]\}\_a((x))\; \backslash quad\; \backslash text\{for\; all\}\; \backslash ;\; \backslash ;\; x\; >\; -2,$

where $(x)\; =\; x\; -\; [x]$ denotes the fractional part of {{mvar|x}} and $\backslash exp^\{[x]\}\_a$ is the $[x]$-iterated function of the function $\backslash exp\_a$.

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 $\{\}^xe$ 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 $f:\; (-2,\; +\backslash infty)\backslash rightarrow\; \backslash mathbb\{R\}$ is a continuous function that satisfies:

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

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

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

File:Approximations of 0.5 tetratrated to the x.png

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

: $\{\}^\{x\}a\; \backslash approx\; \backslash begin\{cases\}$

\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}

which is differentiable for all $x\; >\; 0$, but not twice differentiable. For example, $\{\}^\backslash frac\{1\}\{2\}2\; \backslash approx\; 1.45933...$ If $a\; =\; e$ this is the same as the linear approximation.

Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where $\backslash left(a^\backslash frac\{1\}\{n\}\backslash right)^n\; =\; a$. Namely,

: $$

{}^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

.

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.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".

File:Tetration analytic extension.svg

In 2017, it was proved{{cite journal

|author-first1=W.

|author-last1=Paulsen

|author-first2=S.

|author-last2=Cowgill

|title=Solving $F(z+1)\; =\; b^\{F(z)\}$ 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

}} that there exists a unique function $F$ satisfying

$F(z\; +\; 1)\; =\; \backslash exp\backslash bigl(F(z)\backslash bigr)$ (equivalently $F(z+1)\; =\; b^\{F(z)\}$ when $b=e$), with the auxiliary conditions

$F(0)\; =\; 1$, and

$F(z)\; \backslash to\; \backslash xi\_\{\backslash pm\}$ (the attracting/repelling fixed points of the logarithm, roughly $0.318\; \backslash pm\; 1.337\backslash ,\backslash mathrm\{i\}$) as $z\; \backslash to\; \backslash pm\; i\backslash infty$. Moreover, $F$ is holomorphic on all of $\backslash mathbb\{C\}$ except for the cut along the real axis at $z\; \backslash le\; -2$. This construction was first conjectured by Kouznetsov (2009){{cite journal

|author-first=D.

|author-last=Kouznetsov

|title=Solution of $F(z\; +\; 1)\; =\; \backslash exp(F(z))$ in complex $z$-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

}} and rigorously carried out by Kneser in 1950.{{cite journal

|author-first=H.

|author-last=Kneser

|title=Reelle analytische Lösungen der Gleichung $\backslash varphi(\backslash varphi(x))\; =\; e^x$ und verwandter Funktionalgleichungen

|journal=Journal für die reine und angewandte Mathematik

|volume=187

|pages=56–67

|date=1950

|language=de

}} Paulsen & Cowgill’s proof extends Kneser’s original construction to any base $b>e^\{1/e\}\backslash approx1.445$, and subsequent work showed how to allow $b\; \backslash in\; \backslash mathbb\{C\}$ with $|b|>e^\{1/e\}$.{{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

}}

In May 2025, Vey gave a unified, holomorphic extension for arbitrary complex bases $b\backslash in\; \backslash mathbb\{C\}\backslash setminus\backslash \{0,1\backslash \}$ and complex heights $z\backslash in\backslash mathbb\{C\}$ by means of Schröder’s equation. In particular, one constructs a linearizing coordinate near the attracting (or repelling) fixed point of the map $f(w)=b^w$, and then patches together two analytic expansions (one around each fixed point) to produce a single function $F\_\{b\}(z)$ that satisfies

$F\_\{b\}(z+1)=b^\{\backslash ,F\_\{b\}(z)\}$ and $F\_\{b\}(0)=1$ on all of $\backslash mathbb\{C\}$. The key step is to define

$\backslash displaystyle$

\Phi_{b}(w)=\lim_{n\to\infty}\;s^{n}\Bigl(f^{\circ n}(w)-\alpha\Bigr),

where $\backslash alpha$ is a fixed point of $f(w)=b^w$, $s\; =\; f\text{'}(\backslash alpha)$, and $f^\{\backslash circ\; n\}$ denotes $n$-fold iteration. One then solves Schröder’s functional equation

$\backslash Phi\_\{b\}\backslash bigl(b^\{\backslash ,w\}\backslash bigr)\backslash ;=\backslash ;s\backslash ;\backslash Phi\_\{b\}(w)$

locally (for $w$ near $\backslash alpha$), 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 $a\_\{n\}^\{(b)\}$ in the local Schröder expansion:

$\backslash Phi\_\{b\}(w)$

= \sum_{n=0}^{\infty} a_{n}^{(b)}\,(w-\alpha)^{n},

and gives rigorous bounds proving factorial convergence of $a\_\{n\}^\{(b)\}$.{{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

}}

Using Kneser’s (and Vey’s) tetration, example values include

$\{\}^\{\backslash tfrac\{\backslash pi\}\{2\}\}e\; \backslash approx\; 5.82366\backslash ldots$,

$\{\}^\{\backslash tfrac\{1\}\{2\}\}2\; \backslash approx\; 1.45878\backslash ldots$, and

$\{\}^\{\backslash tfrac\{1\}\{2\}\}e\; \backslash approx\; 1.64635\backslash ldots$.

The requirement that tetration be holomorphic on all of $\backslash mathbb\{C\}$ (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 $\backslash \{\backslash alpha\_\{n\}\backslash \}$ and $\backslash \{\backslash beta\_\{n\}\backslash \}$, one can set

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),

which still satisfies $S(z+1)=b^\{S(z)\}$ and $S(0)=1$, but has additional singularities creeping in from the imaginary direction.

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 )

Tetration can be defined for ordinal numbers via transfinite induction. For all {{math|α}} and all {{math|β > 0}}:

$$\{\}^0\backslash alpha\; =\; 1$$

Tetration (restricted to $\backslash mathbb\{N\}^2$) is not an elementary recursive function. One can prove by induction that for every elementary recursive function {{mvar|f}}, there is a constant {{mvar|c}} such that

: $f(x)\; \backslash leq\; \backslash underbrace\{2^\{2^\{\backslash cdot^\{\backslash cdot^\{x\}\}\}\}\}\_c.$

We denote the right hand side by $g(c,\; x)$. Suppose on the contrary that tetration is elementary recursive. $g(x,\; x)+1$ is also elementary recursive. By the above inequality, there is a constant {{mvar|c}} such that $g(x,x)\; +1\; \backslash leq\; g(c,\; x)$. By letting $x=c$, we have that $g(c,c)\; +\; 1\; \backslash leq\; g(c,\; c)$, a contradiction.

Exponentiation has two inverse operations; roots and logarithms. Analogously, the 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 $\{^3\}y=x$, the two inverses are the cube super-root of {{mvar|y}} and the super-logarithm base {{mvar|y}} of {{mvar|x}}.

{{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 $^n\; y\; =\; x$, then {{mvar|y}} is an {{mvar|n}}th super-root of {{mvar|x}} ($\backslash sqrt[n]\{x\}\_s$ or $\backslash sqrt[4]\{x\}\_s$).

For example,

: $^4\; 2\; =\; 2^\{2^\{2^\{2\}\}\}\; =\; 65\{,\}536$

so 2 is the 4th super-root of 65,536 $\backslash left(\backslash sqrt[4]\{65\{,\}536\}\_s\; =2\backslash right)$.

File:The graph y = √x(s).svg

The 2nd-order super-root, square super-root, or super square root has two equivalent notations, $\backslash mathrm\{ssrt\}(x)$ and $\backslash sqrt\{x\}\_s$. It is the inverse of $^2\; x\; =\; x^x$ and can be represented with the Lambert W function:{{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 |format=PostScript |doi=10.1007/BF02124750 |arxiv=1809.07369 |s2cid=29028411}}

: $\backslash mathrm\{ssrt\}(x)=\backslash exp(W(\backslash ln\; x))=\backslash frac\{\backslash ln\; x\}\{W(\backslash ln\; x)\}$ or

: $\backslash sqrt\{x\}\_s=e^\{W(\backslash ln\; x)\}$

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

: $\backslash sqrt[y]\{x\}\; =\; \backslash log\_y\; x$

Like square roots, 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

At $x\; =\; 1$:

{{center|$$\backslash mathrm\{ssrt\}(x)\; =\; 1\; +\; (x-1)\; -(x-1)^2\; +\; \backslash frac\{3\}\{2\}\; (x-1)^3\; -\; \backslash frac\{17\}\{6\}\; (x-1)^4\; +\; \backslash frac\{37\}\{6\}(x-1)^5\; -\; \backslash frac\{1759\}\{120\}(x-1)^6\; +\; \backslash frac\{13279\}\{360\}\; (x-1)^7\; +\; \backslash mathcal\; O\; \{\backslash left((x-1)^8\; \backslash right)\}$$}}

File:Cube super root.png

One of the simpler and faster formulas for a third-degree super-root is the recursive formula. If $y\; =\; x^\{x^x\}$ then one can use:

• $x\_0\; =\; 1$
• $x\_\{n+1\}\; =\; \backslash exp(W(W(x\_n\backslash ln\; y)))$

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}}, $\backslash sqrt[n]\{x\}\_s$, exists for {{math|x ≥ 1}}.

However, if the linear approximation above is used, then $^y\; x\; =\; y\; +\; 1$ if {{math|−1 < y ≤ 0}}, so $^y\; \backslash sqrt\{y\; +\; 1\}\_s$ cannot exist.

In the same way as the square super-root, terminology for other super-roots can be based on the normal roots: "cube super-roots" can be expressed as $\backslash sqrt[3]\{x\}\_s$; the "4th super-root" can be expressed as $\backslash sqrt[4]\{x\}\_s$; and the "{{mvar|n}}th super-root" is $\backslash sqrt[n]\{x\}\_s$. Note that $\backslash sqrt[n]\{x\}\_s$ 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 $x\; =\; \{^\backslash infty\; y\}\; =\; y^\{\backslash left[^\backslash infty\; y\backslash right]\}\; =\; y^x,$ and thus that $y\; =\; x^\{1/x\}$. Therefore, when it is well defined, $\backslash sqrt[\backslash infty]\{x\}\_s\; =\; x^\{1/x\}$ and, unlike normal tetration, is an elementary function. For example, $\backslash sqrt[\backslash infty]\{2\}\_s\; =\; 2^\{1/2\}\; =\; \backslash sqrt\{2\}$.

It follows from the Gelfond–Schneider theorem that super-root $\backslash sqrt\{n\}\_s$ for any positive integer {{mvar|n}} is either integer or transcendental, and $\backslash sqrt[3]\{n\}\_s$ is either integer or irrational. It is still an open question whether irrational super-roots are transcendental in the latter case.

{{Main|Super-logarithm}}

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

The function {{math|slog_a x}} satisfies:

: $\backslash begin\{align\}$

\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}

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:

• It is not known whether there is an integer $n\; \backslash ge\; 4$ for which {{math|{{sup|n}}π}} is an integer, because we could not calculate precisely enough the numbers of digits after the decimal points of $\backslash pi$.{{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}}{{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}} for $n\; \backslash ge\; 5$, as we are not aware of any other methods besides some direct computation. In fact, since $\backslash log\_\{10\}(e)\; \backslash cdot\; \{\}^\{3\}e\; =\; 1656520.36764$, then $\{\}^\{4\}e\; >\; 2\backslash cdot\; 10^\{1656520\}$. Given and , then for $n\; \backslash ge\; 5$. It is believed that {{math|{{sup|n}}e}} is not an integer for any positive integer {{mvar|n}}, due to the algebraic independence of $e,\; \{\}^\{2\}e,\; \{\}^\{3\}e,\; \backslash dots$, given Schanuel's conjecture.{{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}}
• It is not known whether {{math|{{sup|n}}q}} is rational for any positive integer {{mvar|n}} and positive non-integer rational {{mvar|q}}.[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.] 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}}π}} or {{math|{{sup|π}}e}} (defined using Kneser's extension) are rationals or not.

For each graph H on h vertices and each {{math|ε > 0}}, define

:$D=2\backslash uparrow\backslash uparrow5h^4\backslash log(1/\backslash varepsilon).$

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.Jacob Fox, [https://arxiv.org/abs/1006.1300 A new proof of the graph removal lemma], arXiv preprint (2010). arXiv:1006.1300 [math.CO]

{{Commons category|Tetration|lcfirst=yes}}

{{Wiktionary}}

• Ackermann function
• Big O notation
• Double exponential function
• Hyperoperation
• Iterated logarithm
• Symmetric level-index arithmetic

{{Reflist|group="nb"|refs=

}}

{{Reflist}}

• 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 $y=x^\{[x^\{[x(\backslash cdots)]\}]\}$ für ganzzahliges Argument (Abundanzen)." Mittheilungen der Mathematische Gesellschaft in Hamburg 4, (1901), p. 33–50. (Reference to usage of $\backslash \; \{^\{n\}\; a\}$ 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)

• {{MathWorld |id=PowerTower |title=Power Tower |author=Galidakis, 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