hyper4 tetration
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From: Eleaticus on 10 Sep 2009 12:22 {{distinguish|Titration}} {{Mergefrom|Ultra exponential function|discuss=Talk:Tetration#Ultra exponential|date=August 2008}} {{Mergefrom|Super-root|discuss=Talk:Tetration#Inverse function articles|date=June 2009}} {{Mergefrom|Super-logarithm|discuss=Talk:Tetration#Inverse function articles|date=June 2009}} <!-- NOTE: \,\! in math prevents it from being rendered as HTML, which can't handle nested exponents --> [[Image:TetrationConvergence2D.png|thumbnail|<math>{}^{n}x</math>, for ''n'' > 1, showing convergence to the infinite power tower between the two dots.]] [[Image:InfinitePowerTower.gif|thumb|Infinite power tower.]] In [[mathematics]], '''tetration''' (also known as '''hyper-4''') is an '''iterated exponential''', the first [[hyper operator]] after [[exponentiation]]. The [[portmanteau|portmanteau word]] ''tetration'' was coined by English [[mathematician]] [[Reuben Louis Goodstein]] from [[tetra-]] (four) and [[iterated function|iteration]]. Tetration is used for the [[Large numbers#Standardized system of writing very large numbers|notation of very large numbers]] but has few practical applications, so its study is part of only [[pure mathematics]]. Shown here are examples of the first four hyper operators, with tetration as the fourth: #[[addition]] #:<math> {{a + b} \atop \,} {= \atop \,} {a \, + \atop \, } {{\underbrace{1 + 1 + \cdots + 1}} \atop b}</math> #::1 added to ''a'', ''b'' times. #[[multiplication]] #:<math>{{a \times b = \ } \atop {\ }} {{\underbrace{a + a + \cdots + a}} \atop b}</math> #::''a'' added to itself, ''b'' times. #[[exponentiation]] #:<math>{{a^b = \ } \atop {\ }} {{\underbrace{a \times a \times \cdots \times a}} \atop b}</math> #::''a'' multiplied by itself, ''b'' times. #'''tetration''' #:<math>{\ ^{b}a = \ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^ {a}}}}}} \atop b}</math> #::''a'' exponentiated by itself, ''b'' times. 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 number|complex]] values of <math>~b~</math>, 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 [[Increment|"add one" function]] applied to ''a'', multiplication (''ab'') can be thought of as a chained addition involving ''b'' numbers ''a'', and exponentiation (<math>a^b</math>) can be thought of as a chained multiplication involving ''b'' numbers ''a''. Analogously, tetration (<math>^{b}a</math>) 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 == When evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words: :<math>\,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)} \right)} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536</math> The convention for iterated exponentiation is to work from the right to the left. Thus, :<math>\,\!2^{2^{2^2}} \not= \,\! \left({\left(2^2\right)}^2\right)^2 = 2^{2 \cdot 2 \cdot2} = 256</math>. To generalize the first case (tetration) above, a new notation is needed (see below); however, the second case can be written as :<math>\,\! \left({\left(2^2\right)}^2\right)^2 = 2^{2 \cdot 2 \cdot 2} = 2^{2^3}</math> Thus, ''its'' general form still uses ordinary exponentiation notation. In general, we can use [[Knuth's up-arrow notation]] to write a power as <math>(\uparrow b)(a) = a^b</math> which allows us to write its general form as: :<math>(\uparrow b)^n(a) = a^{b^{n}}</math> == 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 ''[http://links.jstor.org/sici?sici=0022-4812(194712) 12%3A4%3C123%3ATOIRNT%3E2.0.CO%3B2-E Transfinite Ordinals in Recursive Number Theory]'' (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 '''super-exponentiation''' was published by Bromer in his paper ''[http://links.jstor.org/sici?sici=0025-570X(198706) 60%3A3%3C169%3AS%3E2.0.CO%3B2-1 Superexponentiation]'' in 1987. * 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 operator]]s, 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 [http://mathworld.wolfram.com/PowerTower.html power tower] is occasionally used, in the form "the power tower of order ''b''" for <math>{\ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop b} </math> * Ultra exponential is also used, see [[Ultra exponential function]]. Tetration is often confused with closely related functions and expressions. This is because much of the terminology that is used with them can be used with tetration. Here are a few related terms: :{|class="wikitable" ! Form ! Terminology |- |<math>a^{a^{\cdot^{\cdot^{a^a}}}}</math> |Tetration |- |<math>a^{a^{\cdot^{\cdot^{a^x}}}}</math> |Iterated exponentials |- |<math>a_1^{a_2^{\cdot^{\cdot^{a_n}}}}</math> |Nested exponentials (also towers) |- |<math>a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}</math> |Infinite exponentials (also towers) |} In the first two expressions ''a'' is the '''base''', and the number of ''a''s is the '''height''' (add one for ''x''). In the third expression, ''n'' is the '''height''', but each of the bases is different. Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean [[iterated function|iterated]] [[power (mathematics)|power]]s or [[iterated function|iterated]] [[exponential function|exponential]]s. == Notation == The notations in which tetration can be written (some of which allow even higher levels of iteration) include: :{|class="wikitable" ! Name ! Form ! Description |- |Standard notation |<math>\,{}^{b}a</math> |Used by Maurer [1901] and Goodstein[1947]; [[Rudy Rucker]]'s book ''[[Infinity and the Mind]]'' popularized the notation. |- |[[Knuth's up-arrow notation]] |<math>a {\uparrow\uparrow} b</math> |Allows extension by putting more arrows, or, even more powerfully, an indexed arrow. |- |[[Conway chained arrow notation]] |<math>a \rightarrow b \rightarrow 2</math> |Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain |- |[[Ackermann function]] |<math>{}^{b}2 = \operatorname{A}(4, b - 3) + 3</math> |Allows the special case <math>a=2</math> to be written in terms of the [[Ackermann function]]. |- |Iterated exponential notation |<math>{}^{b}a = \exp_a^b(1)</math> |Allows simple extension to iterated exponentials from initial values other than 1. |- | Hooshmand notation<ref name="uxp">{{cite journal |author=M.H.Hooshmand, |year=2006 |title=Ultra power and ultra exponential functions |journal=[[Integral Transforms and Special Functions]] |volume=17 |issue=8 |pages=549â558 |doi= 10.1080/10652460500422247 |url=http://www.informaworld.com/smpp/content~content=a747844256? words=ultra%7cpower%7cultra%7cexponential%7cfunctions&hash=721628008 }} </ref> | <math>\operatorname{uxp}_a b, \, a^{\frac{b}{}} </math> | |- |[[Hyper operator]] notation |<math>a^{(4)}b, \, \operatorname{hyper}_4(a,b)</math> |Allows extension by increasing the number 4; this gives the family of [[hyper operator]]s |- |[[ASCII]] notation |<code>a^^b</code> |Since the up-arrow is used identically to the caret (<code>^</code>), the tetration operator may be written as (<code>^^</code>). |} 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: :<math>\exp_a^n(x) = a^{a^{\cdot^{\cdot^{a^x}}}}</math> with ''n'' ''a'''s. There are not as many notations for iterated exponentials, but here are a few: :{| class="wikitable" ! Name ! Form ! Description |- |Standard notation |<math>\exp_a^n(x)</math> |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. |- |[[Knuth's up-arrow notation]] |<math>(a{\uparrow})^n(x)</math> |Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on [[large numbers]]. |- |Ioannis Galidakis' notation |<math>\,{}^{n}(a, x)</math> |Allows for large expressions in the base; used by Ioannis Galidakis in [http://ioannis.virtualcomposer2000.com/math/papers/Extensions.pdf On Extending hyper4 ... to the Reals]. |- |[[ASCII]] (auxiliary) |<code>a^^n@x</code> |Based on the view that an iterated exponential is ''auxiliary tetration''. |- |[[ASCII]] (standard) |<code>exp_a^n(x)</code> |Based on standard notation. |} == 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. :{| class="wikitable" ! <math>n</math> ! <math>{}^{2}n</math> ! <math>{}^{3}n</math> ! <math>{}^{4}n</math> |- | 1 | 1 | 1 | 1 |- | 2 | 4 | 16 | 65,536 |- | 3 | 27 | 7,625,597,484,987 | <math>\exp_{10}^3(1.09902)</math> |- | 4 | 256 | <math>\exp_{10}^2(2.18788)</math> | <math>\exp_{10}^3(2.18726)</math> |- | 5 | 3,125 | <math>\exp_{10}^2(3.33931)</math> | <math>\exp_{10}^3(3.33928)</math> |- | 6 | 46,656 | <math>\exp_{10}^2(4.55997)</math> | <math>\exp_{10}^3(4.55997)</math> |- | 7 | 823,543 | <math>\exp_{10}^2(5.84259)</math> | <math>\exp_{10}^3(5.84259)</math> |- | 8 | 16,777,216 | <math>\exp_{10}^2(7.18045)</math> | <math>\exp_{10}^3(7.18045)</math> |- | 9 | 387,420,489 | <math>\exp_{10}^2(8.56784)</math> | <math>\exp_{10}^3(8.56784)</math> |- | 10 | 10,000,000,000 | <math>\exp_{10}^3(1)</math> | <math>\exp_{10}^4(1)</math> |} == Extensions == Extending <math>\,{}^{b}x</math> to real numbers <math>x > 0</math> is straightforward and gives, for each natural number <math>b</math>, a '''super-power function''' <math>\,f(x) = {}^{b}x</math>. 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 operator]]s, and ''super'' for the '''rank''' 4 [[hyper operator]], known as tetration. <!--As mentioned above, for positive integers <math>b</math> the function tends to 1 for <math>x</math> tending to 0 if <math>b</math> is even, and to 0 if <math>b</math> is odd, while for <math>b = 0</ math> and <math>b = -1</math> the function is constant, with values 1 and 0, respectively. --> Consider <math>\exp_a^z(t)</math>, where <math>~a~\in \mathcal A</math>, <math>~z~\in \mathcal Z</math>, and <math>~t~\in \mathcal T</math>. Initially, <math>~\mathcal A~</math> may mean "reals", and each of <math>\mathcal Z</math> and <math>\mathcal T</math> may mean "non-negative integers". For the extension to other sets <math>~\mathcal A~</math>, <math>\mathcal Z</math> and <math>\mathcal T</math>, one has no need to deal with a function of 3 variables. Let <math>~F_a(z)=\exp_a^z(1)~</math>, where <math>~F_a~</math> is an invertible function. Then tetration can be expressed as follows: :<math>~\exp_a^z(t)=F_a\!\Big(z+F_a^{-1}(t)\Big)~</math>. For this reason, in the subsections below, various extensions of a function of 2 variables are considered. === Extension to infinitesimal bases === Sometimes, <math>0^0</math> is taken to be an undefined quantity. In this case, values for <math>\,{}^{k}0</math> cannot be defined directly. However, <math>\lim_{n\rightarrow0} {}^{k}n</math> is well defined, and exists: :<math>\lim_{n\rightarrow0} {}^{k}n = \begin{cases} 1, & k \mbox { even} \\ 0, & k \mbox{ odd} \end{cases} </math> This limit holds for negative <math>n</math>, as well. <math>\,{}^{k} 0</math> could be defined in terms of this limit and this would agree with a definition of <math>0^0 = 1</math>. This limit definition holds for <math>{}^{2}0 = 1</math> because 2 is [[Even and odd numbers| even]], and holds for <math>{}^{0}0 = 1</math> because [[0 is even]]. === Extension to complex bases === [[Image:Tetration period.gif|thumbnail|Tetration by period]] [[Image:Tetration escape.gif|thumbnail|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</math>, where <math>i</math> is the [[square root]] of −1. For example, <math> {}^{k}z</math> where <math>z=i</math>, tetration is achieved by using the [[principal branch]] of the natural logarithm, and using [[Euler's formula]] we get the relation: :<math> 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) </math> This suggests a recursive definition for <math>{}^{(k+1)}i = a'+b'i</ math> given any <math>{}^{k}i = a+bi</math>: :<math>a' = e^{-{b\pi \over 2}} \cos{a\pi \over 2}</math> :<math>b' = e^{-{b\pi \over 2}} \sin{a\pi \over 2}</math> The following approximate values can be derived: :{| class="wikitable" ! <math>{}^{k}i</math> ! Approximate Value |- |<math>{}^{1}i = i</math> |''i'' |- |<math>{}^{2}i = i^{\left({}^{1}i\right)}</math> |<math>0.2079</math> |- |<math>{}^{3}i = i^{\left({}^{2}i\right)}</math> |<math>0.9472 + 0.3208i</math> |- |<math>{}^{4}i = i^{\left({}^{3}i\right)}</math> |<math>0.0501 + 0.6021i</math> |- |<math>{}^{5}i = i^{\left({}^{4}i\right)}</math> |<math>0.3872 + 0.0305i</math> |- |<math>{}^{6}i = i^{\left({}^{5}i\right)}</math> |<math>0.7823 + 0.5446i</math> |- |<math>{}^{7}i = i^{\left({}^{6}i\right)}</math> |<math>0.1426 + 0.4005i</math> |- |<math>{}^{8}i = i^{\left({}^{7}i\right)}</math> |<math>0.5198 + 0.1184i</math> |- |<math>{}^{9}i = i^{\left({}^{8}i\right)}</math> |<math>0.5686 + 0.6051i</math> |} Solving the inverse relation as in the previous section, yields the expected <math>\,{}^{0}i = 1</math> and <math>\,{}^{(-1)}i = 0</math>, with negative values of <math>k</math> giving infinite results on the imaginary axis. Plotted in the [[complex plane]], the entire sequence spirals to the limit <math>0.4383 + 0.3606i</math>, which could be interpreted as the value where <math>k</math> 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. === Extension to infinite heights === [[Image:TetrationConvergence3D.png|thumbnail|The function <math>\left | \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} \right |</math> on the complex plane, showing infinite real power towers (black curve)]] Tetration can be extended to [[infinite]] heights (''b'' in <math>{}^ {b}a</math>). 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}}}}</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}^{1.41}}}}} &= \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}</math> In general, the infinite power tower <math>x^{x^{\cdot^{\cdot}}}</ math>, defined as the limit of <math>{}^{n}x</math> as n goes to infinity, converges for ''e''<sup>−''e''</ sup> ⤠''x'' ⤠''e''<sup>1/''e''</sup>, roughly the interval from 0.066 to 1.44, a result shown by [[Leonhard Euler]]. The limit, should it exist, is a positive real solution of the equation ''y'' = ''x''<sup>''y''</sup>. Thus, ''x'' = ''y''<sup>1/''y''</sup>. The limit defining the infinite tetration of ''x'' fails to converge for ''x'' > ''e''<sup>1/''e''</sup> because the maximum of ''y''<sup>1/''y''</sup> is ''e''<sup>1/''e''</sup>. This may be extended to complex numbers ''z'' with the definition: :<math>{}^{\infty}z = z^{z^{\cdot^{\cdot}}} = -\frac{\mathrm{W}(-\ln {z})}{\ln{z}} </math> where W(''z'') represents [[Lambert's W function]]. As the limit <math>y={}^{\infty}x</math> (if existent, i.e. for ''e''<sup>−''e''</ sup> < ''x'' < ''e''<sup>1/''e''</sup>) must satisfy ''x''<sup>''y''</sup> = ''y'' we see that <math>h (x):={}^{\infty}x</math> is (the lower branch of) the inverse function of <math>y\mapsto y^{1/y}</math>. === Extension to negative heights === Tetration can be extended to heights that are negative. Using the relation: :<math>{}^{k}n = \log_n \left({}^{(k+1)}n\right)</math> (which follows from the definition of tetration), one can derive (or define) values for <math>{}^{k}n</math> where <math>k \in \{-1, 0, 1\} </math>. <!-- This is so much easier to read, consistently displayed, and cleaner. Can we keep it? Huh? Huh? Can we? Please? --> :<math> \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} </math> This confirms the intuitive definition of <math>{}^{1}n</math> as simply being <math>n</math>. However, no further values can be derived by further iteration in this fashion, as <math>\log_n 0</math> is undefined. Similarly, since <math>\log_{1} 1</math> is also undefined: :<math>\log_{1} 1 = \frac{\log_n 1}{\log_n 1} = \frac{0}{0}</math> the derivation above does not hold when <math>n</math> = 1. Therefore, <math>{}^{(-1)}1</math> must remain an undefined quantity as well. (The figure <math>{}^{0}1</math> can safely be defined as 1, however.) === Extension to real heights === [[Image:Real-tetration.png|thumbnail|<math>\,{}^{x}e</math> 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 <math>b</math>, 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''' <math>\,f(x) = {}^{x}a</math> over real <math>x > -2</math> that satisfies *<math>\,{}^{(-1)}a = 0</math> *<math>\,{}^{0}a = 1</math> *<math>\,{}^{b}a = a^{\left({}^{(b-1)}a\right)}</math> for all real ''b'' > -1. *A fourth requirement that is usually one of: :*A '''continuity''' requirement (usually just that <math>{}^{x}a</ math> is continuous in both variables for <math>x > 0</math>). :*A '''differentiability''' requirement (can be once, twice, ''n'' times, or infinitely differentiable in ''x''). :*A '''regularity''' requirement (implying twice differentiable in ''x'') that: ::<math>\left( \frac{d^2}{dx^2}f(x) > 0\right)</math> for all <math> x > 0 </math> 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 <math>\,{}^{x}a</math> is defined for an interval of length one, the whole function easily follows for all <math>x > -2</ math>. A '''linear approximation''' (solution to the continuity requirement, approximation to the differentiability requirement) is given by: :<math>{}^{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}</math> hence: :{| class="wikitable" ! Approximation ! Domain |- |<math>\,{}^{x}a \approx x+1</math> |for <math>-1<x<0</math> |- |<math>\,{}^{x}a \approx a^x</math> |for <math>0<x<1</math> |- |<math>\,{}^{x}a \approx a^{a^{(x-1)}}</math> |for <math>1<x<2</math> |} and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by <math>\ln{a}</math>. A '''quadratic approximation''' (to the differentiability requirement) is given by: :<math>{}^{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}</math> which is differentiable for all <math>x > 0</math>, but not twice differentiable. A cubic approximation, and a method for generalizing to approximations of degree ''n'', are given at <ref name=SolveAnalyt>[http:// tetration.itgo.com/paper.html "Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm" by Andrew Robbins]</ ref>. ===Extension to complex heights=== [[Image:2008analuxpFig1d.jpg|400px|right|thumb|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.]] <!-- The existence of an analytic extension of <math>{}^{z}a</math> to complex values of <math>z</math> is not yet established. For <math>a=e</math>, it could be a solution of the [[functional equation]] <math>F(z+1)=\exp(F(z))</math> with the additional conditions that <math>F(0)=1</math> and <math>F(z)</math> remains finite as <math>z\to\pm{\rm i}\infty</math>. !--> The [[conjecture]] is suggested,<ref name ="MOC09">{{cite journal |author=D.Kouznetsov |title="Solution of <math>F(z+1)=\exp(F(z))</math> in complex <math>z</math>-plane" |journal=[[Mathematics of Computation]] |volume=78 |number=267 |pages=1647â1670 |year=2009 |month=July |url= http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/S0025-5718-09-02188-7.pdf}}</ref> that there exists a unique function ''F'' which is a solution of the equation {{nowrap|1=''F''(''z''+1)=exp(''F''(''z''))}} and satisfies the additional conditions that ''F''(0)=1 and ''F''(''z'') approaches the [[fixed point]]s <!-- <math>L,~L^*</math> !--> of the logarithm (roughly 0.31813150520476413531 ± 1.33723570143068940890''i'') as ''z'' approaches ±''i''â and that ''F'' is [[holomorphic]] in the whole complex ''z''-plane, except the part of the real axis at ''z''â¤−2. This function is shown in the figure at right. The complex double precision approximation of this function is available online <ref name="code">Mathematica code for evaluation and plotting of the tetration and its derivatives. http://en.citizendium.org/wiki/TetrationDerivativesReal.jpg/code </ref>. <!-- This function is not [[entire function|entire]], as there are singularities of <math>F(z)</math> on the real axis at the points <math>z=-2,-3,-4,\ldots</math>.!--> The requirement of [[holomorphism]] of tetration is important for the uniqueness. Many functions <math>S</math> can be constructed as : <math>S(z)=F\!\left(~z~ +\sum_{n=1}^{\infty} \sin(2\pi n z)~ \alpha_n +\sum_{n=1}^{\infty} \Big(1-\cos(2\pi n z) \Big) ~\beta_n \right)</ math> where <math>\alpha</math> and <math>\beta</math> are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of <math>\Im(z)</math>. The function ''S'' satisfies the tetration equations {{nowrap| 1=''S''(''z''+1)=exp(''S''(''z''))}}, ''S''(0)=1, and if ''α<sub>n</ sub>'' and ''β<sub>n</sub>'' approach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of {''α''} or {''β''} are not zero, then function ''S'' has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients {''α''} and {''β''} are, the further away these singularities are from the real axis. The extension of tetration into the complex plane is thus essential for the uniqueness; the [[real-analytic]] tetration is not unique. == Super-exponential growth == A super-exponential function grows even faster than a [[double exponential function]]; for example, if <math>a</math> = 10 and we use the (not very smooth) linear approximation: * <math>f(-1)=0</math> * <math>f(0)=1</math> * <math>f(1)=10</math> * <math>f(2)=10^{10}</math> * <math>f(2.3)=10^{100}</math> ([[googol]]) * <math>\,\!f(3)=10^{10^{10}}</math> * <math>\,\!f(3.3)=10^{10^{100}}</math> ([[googolplex]]) * It passes <math>\,\!10^{10^x}</math> at <math>x = 2.376</math>: <math>f(x) \approx 4.83 \times 10^{237}</math> == Inverse functions == The [[inverse function]]s of tetration are called the '''[[super- root]]''' (or hyper-4-root), and the '''[[super-logarithm]]''' (or hyper-4-logarithm). The square super root <math>\mathrm{ssrt}(x)</ math> which is the inverse function of <math>x^x</math> can be represented with the Lambert W function: :<math>\mathrm{ssqrt}(x)=e^{W(\mathrm{ln}(x))}=\frac{\mathrm{ln}(x)}{W (\mathrm{ln}(x))}</math> For each integer {{nowrap|''n'' > 2}}, the function ''<sup>n</sup>x'' is defined and increasing for {{nowrap|''x'' ≥ 1}}, and {{nowrap| 1=<sup>''n''</sup>1 = 1}}, so that the ''n''th super-root of ''x'' exists for {{nowrap|''x'' ≥ 1}}. However, if the linear approximation above is used, then {{nowrap|1= ''<sup>y</sup>x'' = ''y'' + 1}} if -1â¤''y''â¤0, so {{nowrap| 1=<sup>''y''</sup>sroot(''y'' + 1)}} cannot exist. Once a continuous increasing (in ''b'') definition of tetration <sup>''b''</sup>''a'' is selected, the corresponding super-logarithm {{nowrap|slog<sub>''a''</sub> ''b''}} is defined for all real numbers ''b'', and {{nowrap|''a'' > 1}}. The function <math>\mathrm{slog}_a</math> satisfies: :<math>\mathrm{slog}_a a^b = 1 + \mathrm{slog}_a b</math> :<math>\mathrm{slog}_a b = 1 + \mathrm{slog}_a \log_a b</math> :<math>\mathrm{slog}_a b > -2</math> == See also == * [[Ackermann function]] * [[Hyper operator]]s == References == {{reflist}} * Daniel Geisler, ''[http://www.tetration.org/ tetration.org]'' * Ioannis 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)'' * Ioannis 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.)'' * Ioannis 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]]. "Ãber die Funktion <math>y=x^{[x^{[x(\cdots)]}]}</ math> für ganzzahliges Argument (Abundanzen)." ''Mittheilungen der Mathematische Gesellschaft in Hamburg'' '''4''', (1901), p. 33-50. ''(Reference to usage of <math>\ ^ba</math> from Knobel's paper.)'' * [[Reuben Goodstein|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] * http://en.citizendium.org/wiki/Tetration , tetration at citizendium * {{Mathworld|PowerTower|Power Tower}} [[Category:Exponentials]] [[Category:Binary operations]] [[Category:Large numbers]] [[de:Potenzturm]] [[eo:Supereksponento]] [[fr:Tétration]] [[it:Tetrazione]] [[hu:Tetráció]] [[ja:ããã¬ã¼ã·ã§ã³]] [[ru:ТеÑÑаÑиÑ]] |||||||||||||||||||||||||||||||||||| ©2009 MeAmI |||||||||||||||||| |||||||| http://MeAmI.org |||||||||||||||||||||||||||||||||||| ||||||||||||||||||||||||||||'Search for the People!'|||||||| |