Tetration

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 imes b = } atop { underbrace{a + a + cdots + a atop b}
#exponentiation
#:$$a^b = } atop { underbrace{a imes a imes cdots imes 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)

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::$,!\; ^\{4\}2\; =\; 2^\{2^\{2^2\; =\; 2^\{left(2^\{left(2^2\; ight)\}\; ight)\}\; =\; 2^\{left(2^4\; ight)\}\; =\; 2^\{16\}\; =\; 65,!536$The convention for iterated exponentiation is to work from the right to the left. Thus, :$,!2^\{2^\{2^2\; ot=\; ,!\; left(\{left(2^2\; ight)\}^2\; ight)^2\; =\; 2^\{2\; cdot\; 2\; cdot2\}\; =\; 256$.

To generalize the first case (tetration) above, a new notation is needed (see below); however, the second case can be written as:$,!\; left(\{left(2^2\; ight)\}^2\; ight)^2\; =\; 2^\{2\; cdot\; 2\; cdot\; 2\}\; =\; 2^\{2^3\}$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::$(uparrow\; b)^n(a)\; =\; a^\{b^\{n$

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 " [http://links.jstor.org/sici?sici=0025-570X(198706)60%3A3%3C169%3AS%3E2.0.CO%3B2-1 Superexponentiation] " in 1987, and Goodstein published his paper " [http://links.jstor.org/sici?sici=0022-4812(194712)12%3A4%3C123%3ATOIRNT%3E2.0.CO%3B2-E 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 [http://www.faculty.fairfield.edu/jmac/ther/tower.htm 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 [http://mathworld.wolfram.com/PowerTower.html 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:

:

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.

: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::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.

Extension to complex heights

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 $z\; opm\{\; m\; i\}infty$. 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 $z=-2,-3,-4,ldots$.

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 $x\; =\; 2.376$: $f(x)\; approx\; 4.83\; imes\; 10^\{237\}$

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 $mathrm\{ssrt\}(x)$ 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 $mathrm\{slog\}\_a\; b$ is defined for all positive and negative real numbers.

The function $mathrm\{slog\}\_a$ satisfies::$mathrm\{slog\}\_a\; a^b\; =\; 1\; +\; mathrm\{slog\}\_a\; b$:$mathrm\{slog\}\_a\; b\; =\; 1\; +\; mathrm\{slog\}\_a\; log\_a\; b$:$mathrm\{slog\}\_a\; b\; >\; -2$

See also

* Ackermann function
* Hyper operators
* Super-logarithm

References

*Daniel Geisler, " [http://www.tetration.org/ tetration.org] "
*I.N. Galidakis, " [http://ioannis.virtualcomposer2000.com/math/exponents4.html On extending hyper4 to nonintegers] " (undated, 2006 or earlier) "(A simpler, easier to read review of the next reference)"
*I.N. Galidakis, " [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://home.earthlink.net/~mrob/pub/math/ln-notes1.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.)"
* I.N. Galidakis, " [http://ioannis.virtualcomposer2000.com/math/ 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. [http://mathworld.wolfram.com/PowerTower.html Power Tower]
* Joseph MacDonell, " [http://www.faculty.fairfield.edu/jmac/ther/tower.htm Some Critical Points of the Hyperpower Function] ".
* Dave L. Renfro, " [http://mathforum.org/discuss/sci.math/t/350321 Web pages for infinitely iterated exponentials] " "(Compilation of entries from questions about tetration on sci.math.)"
* Andrew Robbins, " [http://tetration.itgo.com 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).

External links

* [http://tetration.itgo.com/ Andrew Robbins' site on tetration]
* [http://www.tetration.org/ Daniel Geisler's site on tetration]
* [http://math.eretrandre.org/tetrationforum/index.php Tetration Forum]
*