|
|
| Line 1: |
Line 1: |
| {{distinguish|titration}}
| | Gabrielle Straub is what for you can call me although it's not the very feminine of names. Fish keeping is what I provide every week. Managing people is my time frame job now. My house is now for South Carolina. Go to my web to find out more: http://prometeu.net<br><br>Here is my blog post clash of clans cheats ([http://prometeu.net just click the following page]) |
| [[Image:TetrationComplexColor.png|thumb|border|||268px|||Complex plot of [[holomorphic]] tetration <math>{}^{z}e</math>]]
| |
| [[Image:TetrationConvergence2D.png|thumbnail|<math>{}^{n}x</math>, for ''n'' = 1, 2, 3 ..., showing convergence to the infinite power tower between the two dots]]
| |
| [[Image:Infinite power tower.svg|thumb|<math>\textstyle \lim_{n\rightarrow \infty} {}^nx</math> of infinite power tower coverges for the bases <math>\textstyle (e^{-1})^e \le x \le e^{e^{-1}}) </math>]]
| |
| In [[mathematics]], '''tetration''' (or '''hyper-4''') is the next [[Hyperoperation|hyper operator]] after [[exponentiation]], and is defined as iterated exponentiation. The word was coined by [[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]]. Shown here are examples of the first four [[hyper operator]]s, with tetration as the fourth (and [[Successor function|succession]], the unary operation denoted <math>a' = a + 1</math> taking <math>a</math> and yielding the number after <math>a</math>, as the 0th):
| |
| #[[Addition]]
| |
| #:<math>a + n = a\!\underbrace{''{}^{\cdots}{}'}_n</math>
| |
| #::''a'' succeeded ''n'' times.
| |
| #[[Multiplication]]
| |
| #:<math>a \times n = \underbrace{a + a + \cdots + a}_n</math>
| |
| #::''a'' added to itself, ''n'' times.
| |
| #[[Exponentiation]]
| |
| #:<math>a^n = \underbrace{a \times a \times \cdots \times a}_n</math>
| |
| #::''a'' multiplied by itself, ''n'' times.
| |
| #Tetration
| |
| #:<math>{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n</math>
| |
| #::''a'' exponentiated by itself, ''n'' times.
| |
| | |
| where each operation is defined by iterating the previous one (the next operation in the sequence is [[pentation]]). 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 ''n'', while for tetration, no such regular generalization is yet established; and tetration is not considered an [[elementary function]].
| |
| | |
| Addition (<math>a + n</math>) is the most basic operation, multiplication (<math>an</math>) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving ''n'' numbers ''a'', and exponentiation (<math>a^n</math>) can be thought of as a chained multiplication involving ''n'' numbers ''a''. Analogously, tetration (<math>^{n}a</math>) can be thought of as a chained power involving ''n'' numbers ''a''. The parameter ''a'' may be called the base-parameter in the following, while the parameter ''n'' 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).
| |
| | |
| == Definition ==
| |
| For any positive [[real number|real]] <math> a > 0 </math> and non-negative [[integer]] <math> n \ge 0 </math>, we define <math>\,\! {^{n}a} </math> by:
| |
| :<math>{^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} &\text{if }n>0 \end{cases} </math>
| |
| | |
| == Iterated powers vs. iterated bases/exponentiation ==
| |
| As we can see from the definition, 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>
| |
| | |
| Note that exponentiation is not [[associative]], so evaluating the expression in the other order will lead to a different answer:
| |
| :<math>\,\! 2^{2^{2^2}} \ne \left[{\left(2^2\right)}^2\right]^2 = 2^{2 \cdot 2 \cdot2} = 256</math>
| |
| | |
| Thus, the exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as [[right-associative]].
| |
| | |
| When ''a'' and 10 are [[coprime]], we can compute the last ''m'' decimal digits of <math>\,\!\ ^{n}a</math> using [[Euler's theorem]].
| |
| | |
| == Terminology ==
| |
| There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.
| |
| | |
| * The term ''tetration'', introduced by Goodstein in his 1947 paper ''Transfinite Ordinals in Recursive Number Theory''<ref>{{cite journal |author=R. L. Goodstein |title=Transfinite ordinals in recursive number theory |jstor=2266486 |journal=Journal of Symbolic Logic |volume=12 |year=1947 |issue=4 |doi=10.2307/2266486 |pages=123–129}}</ref> (generalizing the recursive base-representation used in [[Goodstein's theorem]] to use higher operations), has gained dominance. It was also popularized in Rudy Rucker's ''[[Infinity and the Mind]]''.
| |
| * The term ''superexponentiation'' was published by Bromer in his paper ''Superexponentiation'' in 1987.<ref>{{cite journal |author=N. Bromer |title=Superexponentiation |journal=Mathematics Magazine |volume=60 |issue=3 |year=1987 |pages=169–174 |jstor=2689566}}</ref> It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
| |
| * The term ''hyperpower''<ref>{{cite journal |author=J. F. MacDonnell |title=Somecritical points of the hyperpower function <math>x^{x^{\dots}}</math> |journal=International Journal of Mathematical Education |year=1989 |volume=20 |issue=2 |pages=297–305 |mr=994348 |url=http://www.faculty.fairfield.edu/jmac/ther/tower.htm}}</ref> is a natural combination of ''hyper'' and ''power'', which aptly describes tetration. The problem lies in the meaning of ''hyper'' with respect to the [[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 misleading, since it is only referring to tetration.
| |
| * The term ''power tower''<ref>{{MathWorld |urlname=PowerTower |title=Power Tower}}</ref> is occasionally used, in the form "the power tower of order ''n''" for <math>{\ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop n}</math>
| |
| <!-- (No such article)
| |
| * 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 times ''a'' appears 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)|powers]] or iterated [[exponential function|exponentials]].
| |
| | |
| == Notation ==
| |
| There are many different notation styles that can be used to express tetration. Some of these styles can be used for higher iterations as well (hyper-5, hyper-6, and so on). {{clarify|date=September 2011}}
| |
| :{|class="wikitable"
| |
| ! Name
| |
| ! Form
| |
| ! Description
| |
| |-
| |
| |Standard notation
| |
| |<math>\,{}^{n}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} n</math>
| |
| |Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
| |
| |-
| |
| |[[Conway chained arrow notation]]
| |
| |<math>a \rightarrow n \rightarrow 2</math>
| |
| |Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
| |
| |-
| |
| |[[Ackermann function]]
| |
| |<math>{}^{n}2 = \operatorname{A}(4, n - 3) + 3</math>
| |
| |Allows the special case <math>a=2</math> to be written in terms of the Ackermann function.
| |
| |-
| |
| |Iterated exponential notation
| |
| |<math>{}^{n}a = \exp_a^n(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=549–558
| |
| |doi= 10.1080/10652460500422247
| |
| }}</ref>
| |
| | <math>\operatorname{uxp}_a n, \, a^{\frac{n}{}} </math>
| |
| |
| |
| |-
| |
| |[[Hyper operator]] notation
| |
| |<math>a^{(4)}n, \, \operatorname{hyper}_4(a,n)</math>
| |
| |Allows extension by increasing the number 4; this gives the family of [[hyper operation]]s
| |
| |-
| |
| |[[ASCII]] notation
| |
| |<code>a^^n</code>
| |
| |Since the up-arrow is used identically to the caret (<code>^</code>), the tetration operator may be written as (<code>^^</code>).
| |
| |-
| |
| |Bowers' array notation
| |
| |{a,b,2}
| |
| |}
| |
| | |
| One notation above uses iterated exponential notation; in general this is defined as follows:
| |
| :<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.<ref>Ioannis Galidakis. [http://ioannis.virtualcomposer2000.com/math/papers/Extensions.pdf On Extending hyper4 and Knuth’s Up-arrow Notation to the Reals].</ref>
| |
| |-
| |
| |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.
| |
| |-
| |
| |J Notation
| |
| |<code>x^^:(n-1)x</code>
| |
| |Repeats the exponentiation. See [[J (programming language)]]<ref>{{cite web|title=Power Verb|url=http://www.jsoftware.com/help/dictionary/d202n.htm|work=J Vocabulary|publisher=J Software|accessdate=28 October 2011}}</ref>
| |
| |}
| |
| | |
| == 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>x</math>
| |
| ! <math>{}^{2}x</math>
| |
| ! <math>{}^{3}x</math>
| |
| ! <math>{}^{4}x</math>
| |
| |- align=right
| |
| | 1
| |
| | 1
| |
| | 1
| |
| | 1
| |
| |- align=right
| |
| | 2
| |
| | 4
| |
| | 16
| |
| | 65,536
| |
| |- align=right
| |
| | 3
| |
| | 27
| |
| | 19,683
| |
| | <math>\exp_{10}^3(1.09902)</math>
| |
| |- align=right
| |
| | 4
| |
| | 256
| |
| | 7,625,597,484,987
| |
| | <math>\exp_{10}^3(2.18726)</math>
| |
| |- align=right
| |
| | 5
| |
| | 3,125
| |
| | <math>\exp_{10}^2(3.33931)</math>
| |
| | <math>\exp_{10}^3(3.33928)</math>
| |
| |- align=right
| |
| | 6
| |
| | 46,656
| |
| | <math>\exp_{10}^2(4.55997)</math>
| |
| | <math>\exp_{10}^3(4.55997)</math>
| |
| |- align=right
| |
| | 7
| |
| | 823,543
| |
| | <math>\exp_{10}^2(5.84259)</math>
| |
| | <math>\exp_{10}^3(5.84259)</math>
| |
| |- align=right
| |
| | 8
| |
| | 16,777,216
| |
| | <math>\exp_{10}^2(7.18045)</math>
| |
| | <math>\exp_{10}^3(7.18045)</math>
| |
| |- align=right
| |
| | 9
| |
| | 387,420,489
| |
| | <math>\exp_{10}^2(8.56784)</math>
| |
| | <math>\exp_{10}^3(8.56784)</math>
| |
| |- align=right
| |
| | 10
| |
| | 10,000,000,000
| |
| | <math>\exp_{10}^3(1)</math>
| |
| | <math>\exp_{10}^4(1)</math>
| |
| |}
| |
| | |
| == Extensions ==
| |
| Tetration can be extended to define <math> {^n 0} </math> and other [[domain of a function|domains]] as well.
| |
| | |
| === Extension of domain for bases ===
| |
| | |
| ==== Extension to base zero ====
| |
| The exponential <math>0^0</math> is not consistently defined. Thus, the tetrations <math>\,{^{n}0}</math> are not clearly defined by the formula given earlier. However, <math>\lim_{x\rightarrow0} {}^{n}x</math> is well defined, and exists:
| |
| :<math>\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases} </math>
| |
| Thus we could consistently define <math>{}^{n}0 = \lim_{x\rightarrow0} {}^{n}x</math>. This is equivalent to defining <math> 0^0 = 1 </math>.
| |
| | |
| Under this extension, <math>{}^{0}0 = 1</math>, so the rule <math> {^{0}a} = 1 </math> from the original definition still holds.
| |
| | |
| ==== Extension to complex bases ====
| |
| [[Image:Tetration period.png|thumbnail|Tetration by period]]
| |
| [[Image:Tetration escape.png|thumbnail|Tetration by escape]]
| |
| | |
| Since [[complex number]]s can be raised to powers, tetration can be applied to ''bases'' of the form <math>\scriptstyle z \;=\; a + bi</math>, where <math>\scriptstyle i^2 \;=\; -1</math>. For example, <math>\scriptstyle {}^{n}z</math> where <math>\scriptstyle 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^{\frac{1}{2}{\pi i} (a+bi)} = e^{-\frac{1}{2}{\pi b}} \left(\cos{\frac{\pi a}{2}} + i \sin{\frac{\pi a}{2}}\right)</math>
| |
| | |
| This suggests a recursive definition for <math>\scriptstyle {}^{(n+1)}i \;=\; a'+b'i</math> given any <math>\scriptstyle {}^{n}i \;=\; a+bi</math>:
| |
| | |
| :<math>\begin{align}
| |
| a' &= e^{-\frac{1}{2}{\pi b}} \cos{\frac{\pi a}{2}} \\
| |
| b' &= e^{-\frac{1}{2}{\pi b}} \sin{\frac{\pi a}{2}}
| |
| \end{align}</math>
| |
| | |
| The following approximate values can be derived:
| |
| :{| class="wikitable"
| |
| ! <math>{}^{n}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>\scriptstyle \,{}^{0}i \;=\; 1</math> and <math>\scriptstyle \,{}^{(-1)}i \;=\; 0</math>, with negative values of ''n'' 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 ''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 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.
| |
| | |
| === Extensions of the domain for (iteration) "heights" ===
| |
| | |
| ==== 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 [[Infinity|infinite]] heights (''n'' in <math>{}^{n}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^{\cdot}}}}}</math> converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:
| |
| | |
| :<math>\begin{align}
| |
| \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.414}}}}} &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.63}}}} \\
| |
| &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.76}}} \\
| |
| &\approx \sqrt{2}^{\sqrt{2}^{1.84}} \\
| |
| &\approx \sqrt{2}^{1.89} \\
| |
| &\approx 1.93
| |
| \end{align}</math>
| |
| | |
| In general, the infinite power tower <math>x^{x^{\cdot^{\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]].<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref> The limit, should it exist, is a positive real solution of the equation ''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^{\cdot}}}} = \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} ~,</math>
| |
| where ''W''(''z'') represents [[Lambert's W function]].
| |
| | |
| As the limit ''y'' = <sup>∞</sup>''x'' (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 ''x'' ↦ ''y'' = <sup>∞</sup>''x'' is (the lower branch of) the inverse function of ''y'' ↦ ''x'' = ''y''<sup>1/''y''</sup>.
| |
| | |
| ==== (Limited) extension to negative heights ====
| |
| In order to preserve the original rule:
| |
| :<math> {^{(k+1)}a} = a^{({^{k}a})} </math>
| |
| for negative values of <math>k</math> we must use the recursive relation:
| |
| :<math> {^{k}a} = \log_a \left( {^{(k+1)}a} \right) </math>
| |
| | |
| Thus:
| |
| :<math> {}^{(-1)}a = \log_{a} \left( {}^{0}a \right) = \log_{a} 1 = 0 </math>
| |
| | |
| However smaller negative values cannot be well defined in this way because
| |
| :<math> {}^{(-2)}a = \log_{a} \left( {}^{-1}a \right) = \log_{a} 0 </math>
| |
| which is not well defined.
| |
| | |
| Note further that for <math> n = 1 </math> any definition of <math>\,\! {^{(-1)}1} </math> is consistent with the rule because
| |
| :<math> {^{0}1} = 1 = 1^n </math> for any <math>\,\! n = {^{(-1)}1} </math>.
| |
| | |
| ==== Extension to real heights ====
| |
| | |
| At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of <math>n</math>. Various approaches are mentioned below.
| |
| | |
| In general the problem is finding, for any real ''a'' > 0, a ''super-exponential function'' <math>\,f(x) = {}^{x}a</math> over real ''x'' > −2 that satisfies
| |
| *<math>\,{}^{(-1)}a = 0</math>
| |
| *<math>\,{}^{0}a = 1</math>
| |
| *<math>\,{}^{x}a = a^{\left({}^{(x-1)}a\right)}\text{ for all real }x>-1.</math>
| |
| *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, ''k'' 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 ''x'' > −2.
| |
| | |
| ===== Linear approximation for the extension to real heights =====
| |
| [[Image:Real-tetration.png|thumbnail|<math>\,{}^{x}e</math> using linear approximation.]]
| |
| A ''linear approximation'' (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
| |
| :<math>{}^{x}a \approx \begin{cases}
| |
| \log_a(^{x+1}a) & x \le -1 \\
| |
| 1 + x & -1 < x \le 0 \\
| |
| a^{\left(^{x-1}a\right)} & 0 < x
| |
| \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>.
| |
| | |
| ======Examples======
| |
| <math>\begin{align}
| |
| {}^{\frac{1}{2}\pi}e &\approx 5.868...,\\
| |
| {}^{-4.3}0.5 &\approx 4.03335...
| |
| \end{align}</math>
| |
| | |
| A main theorem in Hooshmand's paper<ref name="uxp"/> states: Let <math> 0 <a \neq 1</math>. If <math>f:(-2,+\infty)\rightarrow \mathbb{R}</math> is continuous and satisfies the conditions:
| |
| | |
| *<math> f(x)=a^{f(x-1)} \; \; \text{for all} \; \; x>-1, \; f(0)=1,</math>
| |
| *<math>f</math> is differentiable on <math>(-1, 0),</math>
| |
| *<math>f^\prime</math> is a nondecreasing or nonincreasing function on <math> (-1,0),</math>
| |
| *<math>f^\prime (0^+) = (\ln a) f^\prime (0^-) \text{ or } f^\prime (-1^+) = f^\prime (0^-).</math>
| |
| | |
| then <math>f</math> is uniquely determined through the equation
| |
| :<math>f(x)=\exp^{[x]}_a (a^{x})=\exp^{[x+1]}_a(x) \quad \text{for all} \; \; x > -2,</math>
| |
| | |
| where <math> (x)=x-[x] </math> denotes the fractional part of x and <math> \exp^{[x]}_a </math> is the <math> [x] </math>-[[iterated function]] of the function <math> \exp_a </math>.
| |
| | |
| The proof is that the second through fourth conditions trivially imply that f is a linear function on [−1, 0].
| |
| | |
| The linear approximation to natural tetration function <math>{}^xe</math> is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:
| |
| | |
| If <math> f: (-2, +\infty)\rightarrow \mathbb{R}</math>
| |
| is a continuous function that satisfies:
| |
| | |
| *<math> f(x)=e^{f(x-1)} \; \; \text{for all} \; \; x > -1, \; f(0)=1,</math>
| |
| *<math>f</math> is convex on <math>(-1,0),</math>
| |
| *<math>f^\prime (0^-)\leq f^\prime (0^+).</math>
| |
| | |
| then <math>f=\text{uxp}</math>. [Here <math>f=\text{uxp}</math> is Hooshmand's name for the linear approximation to the natural tetration function.]
| |
| | |
| The proof is much the same as before; the recursion equation ensures that <math>f^\prime (-1^+) = f^\prime (0^+),</math> and then the convexity condition implies that <math>f</math> is linear on (−1, 0).
| |
| | |
| Therefore the linear approximation to natural tetration is the only solution of the equation <math> f(x)=e^{f(x-1)} \; \; (x>-1)</math> and <math>f(0)=1</math> which is [[convex function|convex]] on <math>(-1,+\infty)</math>. All other sufficiently-differentiable solutions must have an [[inflection point]] on the interval (−1, 0).
| |
| | |
| ===== Higher order approximations for the extension to real heights =====
| |
| 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\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)} & 0 < x
| |
| \end{cases}</math>
| |
| which is differentiable for all <math>x > 0</math>, but not twice differentiable. If <math>a = e</math> this is the same as the linear approximation.
| |
| | |
| Note that this function does not satisfy condition that tetration "cancels out" (for example as in raising to power: <math>(a^{1/n})^{n}=a</math>), because it is calculated top-down (as explained in section [[Tetration#Iterated powers|Iterated powers]] above) namely:
| |
| :<math>{}^{n}({}^{1/n}a)=\underbrace{({}^{1/n}a)^{({}^{1/n}a)^{\cdot^{\cdot^{\cdot^{\cdot^{({}^{1/n}a)}}}}}}}_n\neq a</math>.
| |
| | |
| A cubic approximation and a method for generalizing to approximations of degree ''n'' are given at.<ref name=SolveAnalyt>Andrew Robbins. [http://web.archive.org/web/20090201164821/http://tetration.itgo.com/paper.html Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm].</ref>
| |
| | |
| ==== Extension to complex heights ====
| |
| [[File:Tetration analytic extension.svg|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>.
| |
| !-->
| |
| | |
| There is a [[conjecture]]<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
| |
| |issue=267
| |
| |pages=1647–1670
| |
| |date=July 2009
| |
| |url=http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/S0025-5718-09-02188-7.pdf
| |
| |doi=10.1090/S0025-5718-09-02188-7}}</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 (mathematics)|fixed points]] <!-- <math>L,~L^*</math> !--> of the logarithm (roughly 0.31813150520476413531 ± 1.33723570143068940890''i'')
| |
| as ''z'' approaches ±''i''∞ and that ''F'' is [[holomorphic]] in the whole complex ''z''-plane, except the part of the real axis at ''z''≤−2.
| |
| This function is shown in the figure at right.
| |
| The complex double precision approximation of this function is available online.<ref name="code">[http://en.citizendium.org/wiki/TetrationDerivativesReal.jpg/code Mathematica code for evaluation and plotting of the tetration and its derivatives].</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.
| |
| | |
| == Open questions ==
| |
| * It is not known if <sup>''n''</sup>[[Pi|{{pi}}]] or <sup>''n''</sup>[[E (mathematical constant)|{{math|''e''}}]] is an integer for any positive integer ''n''. Particularly, it is not known if <sup>4</sup>{{pi}} is an integer.
| |
| * It is not known if <sup>''n''</sup>''q'' is an integer for any positive integer ''n'' and positive non-integer rational ''q''.<ref>[http://condor.depaul.edu/mash/atotheamg.pdf Marshall, Ash J., and Tan, Yiren, "A rational number of the form ''a''<sup>''a''</sup> with ''a'' irrational", Mathematical Gazette 96, March 2012, pp. 106-109.]</ref> Particularly, it is not known if the positive root of the equation <sup>4</sup>''x'' = 2 is a rational number.
| |
| | |
| == Inverse relations ==
| |
| [[Exponentiation]] has two inverse relations; [[Nth root|roots]] and [[logarithms]]. Analogously, the [[Inverse function|inverse relation]]s of tetration are often called the ''super-root'', and the ''super-logarithm''.
| |
| | |
| ===Super-root===
| |
| | |
| The super-root is the inverse relation of tetration with respect to the base: if <math>^n y = x</math>, then ''y'' is an ''n''th super root of ''x''.
| |
| For example,
| |
| :<math>^4 2 = 2^{2^{2^{2}}} = 65,536</math>
| |
| | |
| so 2 is the 4th super-root of 65,536 and
| |
| :<math>^3 3 = 3^{3^{3}} = 7,625,597,484,987</math>
| |
| | |
| so 3 is the 3rd super-root (or super cube root) of 7,625,597,484,987.
| |
| | |
| ==== Square super-root ====
| |
| [[Image:The graph y = √x(s).png|thumb|right|The graph y = <math>\sqrt{x}_s</math>.]]
| |
| The ''2nd-order super-root'', ''square super-root'', or ''super square root'' has two equivalent notations, <math>\mathrm{ssrt}(x)</math> and <math>\sqrt{x}_s</math>. It is the inverse of <math>^2 x = x^x</math> and can be represented with the [[Lambert W function]]:<ref name = "Corless">
| |
| {{cite journal
| |
| | last1 = Corless | first1 = R. M.
| |
| | last2 = Gonnet | first2 = G. H.
| |
| | last3 = Hare | first3 = D. E. G.
| |
| | last4 = Jeffrey | first4 = D. J.
| |
| | last5 = Knuth | first5 = D. E.
| |
| | author5-link = Donald Knuth
| |
| | title = On the Lambert W function
| |
| | journal = Advances in Computational Mathematics
| |
| | volume = 5
| |
| | page = 333
| |
| | year = 1996
| |
| | url = http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps <!-- or http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf -->
| |
| | format = [[PostScript]] <!-- or PDF -->
| |
| | doi = 10.1007/BF02124750
| |
| }}</ref>
| |
| | |
| :<math>\mathrm{ssrt}(x)=e^{W(\mathrm{ln}(x))}=\frac{\mathrm{ln}(x)}{W(\mathrm{ln}(x))}</math>
| |
| | |
| The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when <math>y = \mathrm{ssrt}(x)</math>:
| |
| | |
| : <math>\sqrt[y]{x} = \log_y x</math>
| |
| | |
| Like [[square root]]s, the square super-root of ''x'' may not have a single solution. Unlike square roots, determining the number of square super-roots of ''x'' may be difficult. In general, if <math>e^{-1/e}<x<1</math>, then ''x'' has two positive square super-roots between 0 and 1; and if <math>x > 1</math>, then ''x'' has one positive square super-root greater than 1. If ''x'' is positive and less than <math>e^{-1/e}</math> it doesn't have any [[real number|real]] square super-roots, but the formula given above yields countably infinitely many [[complex number|complex]] ones for any finite ''x'' not equal to 1.<ref name = "Corless" /> The function has been used to determine the size of [[data cluster]]s.<ref>[http://webcache.googleusercontent.com/search?q=cache:r1eNe1n4ZA8J:citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.10.8594%26rep%3Drep1%26type%3Dps+%221.559610%22+nasa&cd=8&hl=en&ct=clnk&gl=uk BOSTON UNIVERSITY COLLEGE OF ENGINEERING – EFFICIENT SELF-ORGANIZATION OF LARGE WIRELESS SENSOR NETWORKS]</ref>
| |
| | |
| ==== Other super-roots ====
| |
| 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'', <math>\sqrt[n]{x}_s</math>, exists for {{nowrap|''x'' ≥ 1}}.
| |
| | |
| However, if the [[Tetration#Linear approximation for the extension to real heights|linear approximation above]] is used, then {{nowrap|<math> ^y x = y + 1</math>}} if −1 < ''y'' ≤ 0, so {{nowrap|<math> ^y \sqrt{y + 1}_s </math>}} cannot exist.
| |
| | |
| Other super-roots are expressible under the same basis{{Clarify|date=April 2011}} used with [[nth root|normal roots]]: super cube roots, the function{{dubious|date=April 2011}} that produces ''y'' when <math>x = y^{y^y}</math>, can be expressed as <math>\sqrt[3]{x}_s</math>; the 4th super-root can be expressed as <math>\sqrt[4]{x}_s</math>; and it can therefore be said that the ''n''<sup>th</sup> super-root is <math>\sqrt[n]{x}_s</math>. Note that <math>\sqrt[n]{x}_s</math> may not be uniquely defined, because there may be more than one ''n''<sup>th</sup> root. For example, ''x'' has a single (real) super-root if ''n'' is ''odd'', and up to two if ''n'' is ''even''.{{Citation needed|date=October 2009}}
| |
| | |
| The super-root can be extended to <math> n = \infty </math>, and this shows a link to the mathematical constant ''[[e (mathematical constant)|e]]'' as it is only well-defined if 1/''e'' ≤ ''x'' ≤ ''e'' (see [[Tetration#Extension_to_infinite_heights|extension of tetration to infinite heights]]). Note that <math> x = {^\infty y} </math> implies that <math> x = y^x </math> and thus that <math> y = x^{1/x} </math>. Therefore, when it is well defined, <math> \sqrt[\infty]{x}_s = x^{1/x} </math> and thus it is an [[elementary function]]. For example, <math>\sqrt[\infty]{2}_s = 2^{1/2} = \sqrt{2}</math>.
| |
| | |
| It follows from the [[Gelfond–Schneider theorem]] that super-root <math>\sqrt[2]{n}_s</math> for any positive integer ''n'' is either integer or [[Transcendental number|transcendental]], and <math>\sqrt[3]{n}_s</math> is either integer or irrational.<ref>[http://condor.depaul.edu/mash/atotheamg.pdf Marshall, Ash J., and Tan, Yiren, "A rational number of the form ''a''<sup>''a''</sup> with ''a'' irrational", Mathematical Gazette 96, March 2012, pp. 106-109.]</ref> But it is still an open question whether irrational super-roots are transcendental in the latter case.
| |
| | |
| ===Super-logarithm===
| |
| {{main|Super-logarithm}}
| |
| Once a continuous increasing (in ''x'') definition of tetration, <sup>''x''</sup>''a'', is selected, the corresponding super-logarithm {{nowrap|slog<sub>''a''</sub> ''x''}} is defined for all real numbers ''x'', and {{nowrap|''a'' > 1}}.
| |
| | |
| The function <math>\mathrm{slog}_a</math> satisfies:
| |
| :<math>\mathrm{slog}_a {^x a} = x</math>
| |
| :<math>\mathrm{slog}_a a^x = 1 + \mathrm{slog}_a x</math>
| |
| :<math>\mathrm{slog}_a x = 1 + \mathrm{slog}_a \log_a x</math>
| |
| :<math>\mathrm{slog}_a x > -2</math>
| |
| | |
| == See also ==
| |
| * [[Ackermann function]]
| |
| * [[Double exponential function]]
| |
| * [[Hyperoperation]]
| |
| * [[Iterated logarithm]]
| |
| * [[Symmetric level-index arithmetic]]
| |
| | |
| == References ==
| |
| {{reflist|2}}
| |
| | |
| * Daniel Geisler, ''[http://www.tetration.org/ tetration.org]''
| |
| * Ioannis Galidakis, ''[http://web.archive.org/web/20090520164620/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://web.archive.org/web/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, ''[http://web.archive.org/web/20090420132250/http://ioannis.virtualcomposer2000.com/math/index.html 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.)''
| |
| * R. Knobel. "Exponentials Reiterated." ''[[American Mathematical Monthly]]'' '''88''', (1981), p. 235–252.
| |
| * Hans Maurer. "Über die Funktion <math>y=x^{[x^{[x(\cdots)]}]}</math> für ganzzahliges Argument (Abundanzen)." ''Mittheilungen der Mathematische Gesellschaft in Hamburg'' '''4''', (1901), p. 33–50. ''(Reference to usage of <math>\ {^{n} a}</math> from Knobel's paper.)''
| |
| * Ripà, Marco (2011). ''La strana coda della serie n^n^...^n'', Trento, UNI Service. ISBN 978-88-6178-789-6
| |
| | |
| ==External links==
| |
| * [http://www.tetration.org/ Daniel Geisler's site on tetration]
| |
| * [http://math.eretrandre.org/tetrationforum/index.php Tetration Forum]
| |
| * [http://go.helms-net.de/math/tetdocs/ Gottfried Helms' site on tetration]
| |
| | |
| [[Category:Exponentials]]
| |
| [[Category:Binary operations]]
| |
| [[Category:Large numbers]]
| |