Tetration

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Not to be confused with Titration.

n x

, for n > 1, showing convergence to the infinite power tower between the two dots.

Infinite power tower.

In mathematics, tetration (also known as hyper-4) is an iterated exponential, the first hyper operator after exponentiation. The portmanteau word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is used for the notation of very large numbers, but has few practical applications so its study is part of pure mathematics. Shown here are examples of the first four hyper operators, with tetration as the fourth:

addition

${{a + b} \atop \,} {= \atop \,} {a \, + \atop \, } {{\underbrace{1 + 1 + \cdots + 1}} \atop b}$
multiplication

${{a \times b = \ } \atop {\ }} {{\underbrace{a + a + \cdots + a}} \atop b}$
exponentiation

${{a^b = \ } \atop {\ }} {{\underbrace{a \times a \times \cdots \times a}} \atop b}$
tetration

${\ ^{b}a = \ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop b}$

where each operation is defined by iterating the previous one. The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of $~b~$ , while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.

Addition (a+b) can be thought of as being b iterations of the "add one" function applied to a, multiplication (ab) can be thought of as a chained addition involving b numbers a, and exponentiation ( $a b$ ) can be thought of as a chained multiplication involving b numbers a. Analogously, tetration ( $b a$ ) can be thought of as a chained power involving b numbers a. The parameter a may be called the base-parameter in the following, while the parameter b in the following may be called the height-parameter (which is integral in the first approach but may be generalized to fractional, real and complex heights, see below)

1 Iterated powers
2 Terminology
3 Notation
4 Examples
5 Extensions
6 Super-exponential growth
7 Approaches to inverse functions
8 See also
9 References
10 External links

[edit] Iterated powers

Note that when evaluating multiple-level exponentiation, the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:

$\,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536$

The convention for iterated exponentiation is to work from the right to the left. Thus,

$\,\!2^{2^{2^2}} \not= \,\! \left({\left(2^2\right)}^2\right)^2 = 2^{2 \cdot 2 \cdot2} = 256$ .

To generalize the first case (tetration) above, a new notation is needed (see below); however, the second case can be written as

$\,\! \left({\left(2^2\right)}^2\right)^2 = 2^{2 \cdot 2 \cdot 2} = 2^{2^3}$

Thus, its general form still uses ordinary exponentiation notation.

In general, we can use Knuth's up-arrow notation to write a power as $(\uparrow b)(a) = a^b$ which allows us to write its general form as:

$(\uparrow b)^n(a) = a^{b^{n}}$

[edit] 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 super-exponentiation is the most proper candidate for a name, however, Bromer published his paper Superexponentiation in 1987, and Goodstein published his paper Transfinite Ordinals in Recursive Number Theory (which coined the term tetration) in 1947, which predates Bromer. So although this is not a misnomer, the shorter and older term has gained more use.
The term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyper operator hierarchy. When considering hyper operators, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is a misnomer, since it is only referring to tetration.
The term power tower has a nice ring to it, but it is a misnomer in two different manners. The first way is that tetration has nothing to do with power functions, as it is a special case of iterated exponentials (see the section iterated powers above). The second way is that the term tower is used to describe nested exponentials which are much more general expressions than tetration. Since power tower is a misnomer in more than one way, it has fallen out of use.
Ultra exponential is also used, see Ultra exponential function.

Tetration is often confused with closely related functions and expressions. This is because many of the terminology that is used with them can be used with tetration. Here are a few related terms:

Form	Terminology
$a^{a^{\cdot^{\cdot^{a^a}}}}$	Tetration
$a^{a^{\cdot^{\cdot^{a^x}}}}$	Iterated exponentials
$a_1^{a_2^{\cdot^{\cdot^{a_n}}}}$	Nested exponentials (also towers)
$a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}$	Infinite exponentials (also towers)

In the first two expressions a is the base, and the number of as is the height (add one for x). In the third expression, n is the height, but each of the bases are 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.

[edit] Notation

The notations in which tetration can be written (some of which allow even higher levels of iteration) include:

Name	Form	Description
Standard notation	$\,{}^{b}a$	Used by Goodstein[1947]; Rudy Rucker's book Infinity and the Mind popularized the notation.
Knuth's up-arrow notation	$a {\uparrow\uparrow} b$	Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
Conway chained arrow notation	$a \rightarrow b \rightarrow 2$	Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
Ackermann function	${}^{b}2 = \operatorname{A}(4, b - 3) + 3$	Allows the special case $a = 2$ to be written in terms of the Ackermann function.
Iterated exponential notation	${}^{b}a = \exp_a^b(1)$	Allows simple extension to iterated exponentials from initial values other than 1.
Hooshmand notation^[1]	$\operatorname{uxp}_a b, \, a^{\frac{b}{}}$
Hyper operator notation	$a^{(4)}b, \, \operatorname{hyper}_4(a,b)$	Allows extension by increasing the number 4; this gives the family of hyper operators
ASCII notation	`a^^b`	Since the up-arrow is used identically to the caret (`^`), the tetration operator may be written as (`^^`).

One notation above shows that tetration can be written as an iterated exponential function where the initial value is one. As a reminder, iterated exponentials have the general form:

$\exp_a^n(x) = a^{a^{\cdot^{\cdot^{a^x}}}}$ with n a's.

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

Name	Form	Description
Standard notation	$\exp_a^n(x)$	Euler coined the notation $exp a (x) = a x$ , and iteration notation $f n (x)$ has been around about as long.
Knuth's up-arrow notation	$(a{\uparrow})^n(x)$	Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on large numbers.
Galidakis' notation	$\,{}^{n}(a, x)$	Allows for large expressions in the base; used by Ioannis Galidakis in On Extending hyper4 ... to the Reals.
ASCII (auxiliary)	`a^^n@x`	Based on the view that an iterated exponential is auxiliary tetration.
ASCII (standard)	`exp_a^n(x)`	Based on standard notation.

[edit] Examples

In the following table, most values are too large to write in scientific notation, so iterated exponential notation is employed to express them in base 10. The values containing a decimal point are approximate.

$n$	$2 n$	$3 n$	$4 n$
0	1	0	1
1	1	1	1
2	4	16	65,536
3	27	7,625,597,484,987	$\exp_{10}^3(1.09902)$
4	256	$\exp_{10}^2(2.18788)$	$\exp_{10}^3(2.18726)$
5	3,125	$\exp_{10}^2(3.33931)$	$\exp_{10}^3(3.33928)$
6	46,656	$\exp_{10}^2(4.55997)$	$\exp_{10}^3(4.55997)$
7	823,543	$\exp_{10}^2(5.84259)$	$\exp_{10}^3(5.84259)$
8	16,777,216	$\exp_{10}^2(7.18045)$	$\exp_{10}^3(7.18045)$
9	387,420,489	$\exp_{10}^2(8.56784)$	$\exp_{10}^3(8.56784)$
10	10,000,000,000	$\exp_{10}^3(1)$	$\exp_{10}^4(1)$

[edit] Extensions

Extending $\,{}^{b}x$ to real numbers $x > 0$ is straightforward and gives, for each natural number $b$ , a super-power function $\,f(x) = {}^{b}x$ . The term super is sometimes replaced by hyper, but this only applies to tetration with integer height, and is falling out of usage. All other uses of the two prefixes use the convention: hyper for all ranks of hyper operators, and super for the rank 4 hyper operator, known as tetration.

Consider $\exp_a^z(t)$ , where $~a~\in \mathcal A$ , $~z~\in \mathcal Z$ , and $~t~\in \mathcal T$ . Initially, $~\mathcal A~$ may mean "reals", and each of $\mathcal Z$ and $\mathcal T$ may mean "non-negative integers". For the extension to other sets $~\mathcal A~$ , $\mathcal Z$ and $\mathcal T$ , one has no need to deal with a function of 3 variables. Let $~F_a(z)=\exp_a^z(1)~$ , where $~F_a~$ is an invertible function. Then tetration can be expressed as follows:

$~\exp_a^z(t)=F_a\!\Big(z+F_a^{-1}(t)\Big)~$ .

For this reason, in the subsections below, various extensions of a function of 2 variables are considered.

[edit] Extension to infinitesimal bases

Sometimes, $00$ is taken to be an undefined quantity. In this case, values for $\,{}^{k}0$ cannot be defined directly. However, $\lim_{n\rightarrow0} {}^{k}n$ is well defined, and exists:

$\lim_{n\rightarrow0} {}^{k}n = \begin{cases} 1, & k \mbox{ even} \\ 0, & k \mbox{ odd} \end{cases}$

This limit holds for negative $n$ , as well. $\,{}^{k}0$ could be defined in terms of this limit and this would agree with a definition of $00 = 1$ . This limit definition holds for $2 0 = 1$ because 2 is even, and holds for $0 0 = 1$ because 0 is even.

[edit] Extension to complex bases

Tetration by period

Tetration by escape

Since complex numbers can be raised to powers, tetration can be applied to bases of the form $z = a + b i$ , where $i$ is the square root of −1. For example, $k z$ where $z = i$ , tetration is achieved by using the principal branch of the natural logarithm, and using Euler's formula we get the relation:

$i^{a+bi} = e^{{i\pi \over 2} (a+bi)} = e^{-{b\pi \over 2}} \left(\cos{a\pi \over 2} + i \sin{a\pi \over 2}\right)$

This suggests a recursive definition for $(k + 1) i = a' + b' i$ given any $k i = a + b i$ :

$a' = e^{-{b\pi \over 2}} \cos{a\pi \over 2}$

$b' = e^{-{b\pi \over 2}} \sin{a\pi \over 2}$

The following approximate values can be derived:

$k i$	Approximate Value
$1 i = i$	i
${}^{2}i = i^{\left({}^{1}i\right)}$	$0.2079$
${}^{3}i = i^{\left({}^{2}i\right)}$	$0.9472 + 0.3208 i$
${}^{4}i = i^{\left({}^{3}i\right)}$	$0.0501 + 0.6021 i$
${}^{5}i = i^{\left({}^{4}i\right)}$	$0.3872 + 0.0305 i$
${}^{6}i = i^{\left({}^{5}i\right)}$	$0.7823 + 0.5446 i$
${}^{7}i = i^{\left({}^{6}i\right)}$	$0.1426 + 0.4005 i$
${}^{8}i = i^{\left({}^{7}i\right)}$	$0.5198 + 0.1184 i$
${}^{9}i = i^{\left({}^{8}i\right)}$	$0.5686 + 0.6051 i$

Solving the inverse relation as in the previous section, yields the expected $\,{}^{0}i = 1$ and $\,{}^{(-1)}i = 0$ , with negative values of $k$ giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit $0.4383 + 0.3606 i$ , which could be interpreted as the value where $k$ is infinite.

Such tetration sequences have been studied since the time of Euler but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the power tower function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.

[edit] Extension to infinite heights

The function $\left | \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} \right |$ on the complex plane, showing infinite real power towers (black curve)

Tetration can be extended to heights (b in $b a$ ) that are not finite, but infinite. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity. For example, $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot}}}}$ 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:

$\begin{align} \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.41}}}}} &= \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.63}}}} \\ &= \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.76}}} = \sqrt{2}^{\sqrt{2}^{1.84}} \\ &= \sqrt{2}^{1.89} = 1.93 \end{align}$

In general, the infinite power tower $x^{x^{\cdot^{\cdot}}}$ , defined as the limit of $n x$ as n goes to infinity, converges for e^−e ≤ x ≤ e^1/e, roughly the interval from 0.066 to 1.44, a result due to Leonhard Euler. The limit, should it exist, is a positive real solution of the equation y = x^y. The inverse function is thus x = y^1/y. The limit defining the infinite tetration of x fails to converge for x > e^1/e because the maximum of y^1/y is e^1/e.

This may be extended to complex numbers z with the definition:

${}^{\infty}z = z^{z^{\cdot^{\cdot}}} = -\frac{\mathrm{W}(-\ln{z})}{\ln{z}}$

where W(z) represents Lambert's W function. As the limit $y={}^{\infty}x$ (if existent, i.e. for e^−e < x < e^1/e) must satisfy x^y = y we see that $h(x):={}^{\infty}x$ is (the lower branch of) the inverse function of $y\mapsto y^{1/y}$ .

[edit] Extension to negative heights

Tetration can be extended to heights that are negative. Using the relation:

${}^{k}n = \log_n \left({}^{(k+1)}n\right)$

(which follows from the definition of tetration), one can derive (or define) values for $k n$ where $k \in \{-1, 0, 1\}$ .

$\begin{array}{rclclcccc} {}^{1}n & = & \log_n \left({}^{2}n\right) & = & \log_{n} \left(n^n\right) & = & n \log_{n} n & = & n \\ {}^{0}n & = & \log_{n} \left({}^{1}n\right) & = & \log_{n} n & & & = & 1 \\ {}^{(-1)}n & = & \log_{n} \left({}^{0}n\right) & = & \log_{n} 1 & & & = & 0 \end{array}$

This confirms the intuitive definition of $1 n$ as simply being $n$ . However, no further values can be derived by further iteration in this fashion, as $log n 0$ is undefined.

Similarly, since $log 11$ is also undefined:

$\log_{1} 1 = \frac{\log_n 1}{\log_n 1} = \frac{0}{0}$

the derivation above does not hold when $n$ = 1. Therefore, $( - 1) 1$ must remain an undefined quantity as well. (The figure $01$ can safely be defined as 1, however.)

[edit] Extension to real heights

$\,{}^{x}e$ using linear approximation.

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of $b$ , although it is an active area of research. Various approaches are mentioned below. For an approach that is still disputed until it has been reviewed further, see ultra exponential function.

In general the problem is finding, for any real a > 0, a super-exponential function $\,f(x) = {}^{x}a$ over real $x > - 2$ that satisfies

$\,{}^{(-1)}a = 0$
$\,{}^{0}a = 1$
$\,{}^{b}a = a^{\left({}^{(b-1)}a\right)}$ for all real b > -1.
A fourth requirement that is usually one of:

A continuity requirement (usually just that $x a$ is continuous in both variables for $x > 0$ ).
A differentiability requirement (can be once, twice, n times, or infinitely differentiable in x).
A regularity requirement (implying twice differentiable in x) that:

$\left( \frac{d^2}{dx^2}f(x) > 0\right)$ for all

x > 0

The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights, one is based on the regularity requirement, and one is based on the differentiability requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.

Fortunately, any solution that satisfies one of these in an interval of length one can be extended to a solution for all positive real numbers. When $\,{}^{x}a$ is defined for an interval of length one, the whole function easily follows for all $x > - 2$ .

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

${}^{x}a \approx \begin{cases} \log_a({}^{(x+1)}a) & x \le -1 \\ 1 + x & -1 < x \le 0 \\ a^{\left({}^{(x-1)}a\right)} & x > 0 \end{cases}$

hence:

Approximation	Domain
$\,{}^{x}a \approx x+1$	for $- 1 < x < 0$
$\,{}^{x}a \approx a^x$	for $0 < x < 1$
$\,{}^{x}a \approx a^{a^{(x-1)}}$	for $1 < x < 2$

and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by $ln a$ .

A quadratic approximation (to the differentiability requirement) is given by:

${}^{x}a \approx \begin{cases} \log_a({}^{(x+1)}a) & x \le -1 \\ 1 + \frac{2\log(a)}{1+\log(a)}x - \frac{1-\log(a)}{1+\log(a)}x^2 & -1 < x \le 0 \\ a^{\left({}^{(x-1)}a\right)} & x > 0 \end{cases}$

which is differentiable for all $x > 0$ , but not twice differentiable.

Other, more complicated solutions may be smoother and/or satisfy additional properties. When defining $x a$ for every a, another possible requirement could be that $x a$ is monotonically increasing with a. Other solutions require not just continuity, but differentiability, or even infinite differentiability. Another approach is to define tetration over real heights as the inverse of the super-logarithm, which is its inverse function with respect to the height.

[edit] Extension to complex heights

Drawing of the analytic extension

f = F (x + i y)

of tetration to the complex plane. Levels $|f|=1,e^{\pm 1},e^{\pm 2},\ldots$ and levels $\arg(f)=0,\pm 1,\pm 2,\ldots$ are shown with thick curves.

The existence of an analytic extension of $z a$ to complex values of $z$ is not yet established. For $a = e$ , it could be a solution of the functional equation $F (z + 1) = exp(F (z))$ with the additional conditions that $F (0) = 1$ and $F (z)$ remains finite as $z\to\pm{\rm i}\infty$ . If such an extension exists, it might have the shape shown in the figure at right. This function is not entire, as there are singularities of $F (z)$ on the real axis at the points $z=-2,-3,-4,\ldots$ .

[edit] Super-exponential growth

A super-exponential function grows even faster than a double exponential function; for example, if $a$ = 10:

$f ( - 1) = 0$
$f (0) = 1$
$f (1) = 10$
$f (2) = 10 10$
$f (2.3) = 10 100$ (googol)
$\,\!f(3)=10^{10^{10}}$
$\,\!f(3.3)=10^{10^{100}}$ (googolplex)
It passes $\,\!10^{10^x}$ at $x = 2.376$ : $f(x) \approx 4.83 \times 10^{237}$

[edit] Approaches to inverse functions

The inverse functions of tetration are called the super-root (or hyper-4-root), and the super-logarithm (or hyper-4-logarithm). The square super root $ssrt(x)$ which is the inverse function of $x x$ can be represented with the Lambert W function:

$\mathrm{ssqrt}(x)=e^{W(\mathrm{ln}(x))}=\frac{\mathrm{ln}(x)}{W(\mathrm{ln}(x))}$

The super-logarithm $slog a b$ is defined for all positive and negative real numbers.

The function $slog a$ satisfies:

slog a a b = 1 + slog a b

slog a b = 1 + slog a log a b

slog a b > - 2

[edit] See also

[edit] References

^ M.H.Hooshmand, (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions 17 (8): 549-558. doi:10.1080/10652460500422247, http://www.informaworld.com/smpp/content~content=a747844256?words=ultra%7cpower%7cultra%7cexponential%7cfunctions&hash=721628008.

Daniel Geisler, tetration.org
Ioannis Galidakis, On extending hyper4 to nonintegers (undated, 2006 or earlier) (A simpler, easier to read review of the next reference)
Ioannis Galidakis , On Extending hyper4 and Knuth's Up-arrow Notation to the Reals (undated, 2006 or earlier).
Robert Munafo, Extension of the hyper4 function to reals (An informal discussion about extending tetration to the real numbers.)
Lode Vandevenne, Tetration of the Square Root of Two, (2004). (Attempt to extend tetration to real numbers.)
Ioannis Galidakis, Mathematics, (Definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.)
Galidakis, Ioannis and Weisstein, Eric W. Power Tower
Joseph MacDonell, Some Critical Points of the Hyperpower Function.
Dave L. Renfro, Web pages for infinitely iterated exponentials (Compilation of entries from questions about tetration on sci.math.)
Andrew Robbins, Home of Tetration (An infinitely differentiable extension of tetration to real numbers.)
R. Knobel. "Exponentials Reiterated." American Mathematical Monthly 88, (1981), p. 235-252.
Hans Maurer. "Über die Funktion $y=x^{[x^{[x(\cdots)]}]}$ für ganzzahliges Argument (Abundanzen)." Mittheilungen der Mathematische Gesellschaft in Hamburg 4, (1901), p. 33-50. (Reference to usage of $\ ^ba$ from Knobel's paper.)
Reuben Louis Goodstein. "Transfinite ordinals in recursive number theory." Journal of Symbolic Logic 12, (1947).

[edit] External links

Andrew Robbins' site on tetration
Daniel Geisler's site on tetration
Tetration Forum
http://en.citizendium.org/description/Tetration , tetration at citizendium
Eric W. Weisstein, Power Tower at MathWorld.

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