Difference between revisions of "Tetration"
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\(y=\mathrm{tet}_b(x)\) versus \(x\) for various values of base \(b\). Суперфункции[1],с.244,Рис.17.1; Superfunctions[2],p.239,Fig.17.2.
First five ackermanns. Superfunctions[2],p.266,Fig.19.7.
Map of natural tetration. Superfunctions[2],p.203,Fig.4.12
Complex maps of tetration \(\mathrm{tet}_b\) to base \(b\!=\!1.5\) , left; \(b\!=\!\exp(1/\mathrm e)\) , center ; \(b\!=\!\sqrt{2}\) , right. Суперфункции[1],с.251,Рис.17.4; Superfunctions[2],p.245,Fig.17.4;
Tetration to Sheldon base.
Суперфункции[1],с.257,Рис.18.3; Superfunctions[2],p.250, top of Fig.18.3.
"half-sheep". Суперфункции[1],с.218,Рис.15.6; Superfunctions [2],p.215,Fig.15.6
(shorten; refs) |
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| − | <div style="float:right; margin:-72px 0px |
+ | <div class="thumb tright" style="float:right; margin:-72px 0px 8px 18px; line-height:11px; width:300px"> |
| − | {{pic|Tetreal10bx10d.png| |
+ | {{pic|Tetreal10bx10d.png|300px}}<small><center>\(y=\mathrm{tet}_b(x)\) versus \(x\) for various values of base \(b\). [[Суперфункции]]<ref name="ru"/>,с.244,Рис.17.1; [[Superfunctions]]<ref name="en"/>,p.239,Fig.17.2.</center></small> |
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| − | {{pic|Ackerplot.jpg|320px}}<small><center>Natural [[tetration]] (dashed) and other [[Ackermann function|ackermanns]]</center></small> |
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</div> |
</div> |
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| + | <div class="thumb tright" style="float:right; background-color:#fff; margin:2px 0px 8px 8px; width:300px; line-height:9px"> |
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| − | [[Tetration]] (or Tetrational) \({\rm tet}_b\) to base \(b \in \mathbb R\), \(b\!>1\)<br> |
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| + | {{pic|Ackerplot.jpg|300px}}<small><center>First five [[ackermann]]s. [[Superfunctions]]<ref name="en"> |
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| − | is the \(\mathbb C \mapsto \mathbb C\) function, which is solution of equations |
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| + | https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862<br> |
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| − | : \({\rm tet}_b(z\!+\!1) = \exp_b\!\big( {\rm tet}_b(z) \big)\) |
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| + | https://mizugadro.mydns.jp/BOOK/468.pdf |
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| − | : \({\rm tet}_b(0) = 1\) |
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| + | Dmitrii Kouznetsov. Superfunctions. [[Lambert Academic Publishing]], 2020. |
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| − | such that, at least for \(b\!>\!1\), is holomorphic at least in \(\{ z \in \mathbb C : \Re(z)\!>\!-2\}\). |
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| + | </ref>,p.266,Fig.19.7.</center></small> |
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| − | |||
| + | </div> |
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| − | Parameter \(b\) is called "base", as in the cases of [[exponential]] and [[logarithm]]. According to the first of equations above, \(\mathrm{tet}_b\) is [[superfunction]] of [[exponential]] to base \(b\). Exponential is [[transfer function]] of tetration. |
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| + | <div class="thumb tright" style="float:right; margin:8px 0px 14px 12px; background-color:#fff; width:220px; line-height:11px;"> |
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| − | |||
| + | {{pic|B271t.png|240px}}<small><center>Map of [[natural tetration]]. [[Superfunctions]]<ref name="en"/>,p.203,Fig.4.12 </center></small> |
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| − | For real \(b\), it is assumed that \({\rm tet}_b(z^*)={\rm tet}_b(z)^*\), where the asterisk means the complex conjugation. For the case of base \(b \!=\! \mathrm e\), the index may be omitted (as in the case of exp and logarithm), id est, \({\rm tet}(z)={\rm tet}_{\rm e}(z)\). The solution is believed to exist and to be unique, although not all mathematicians agree that the proof of the existence and the uniqueness is rigorous. |
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| + | </div> |
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| + | [[Tetration]] ([[Тетрация]]) is the [[superfunction]] of the [[exponential]] map. |
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| + | For a given base \(b\), the tetration \(\operatorname{tet}_b\) is defined as the function satisfying the transfer equation |
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| + | \[ |
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| + | \operatorname{tet}_b(z+1) = b^{\operatorname{tet}_b(z)} |
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| + | \] |
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| + | together with additional normalization and regularity conditions that select a unique solution among infinitely many possible superfunctions. |
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| + | For real values of the argunet, the explicit plots of \(\operatorname{tet}_b(x)\) versos \(x\) |
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| + | for various real values of base \(b\) is shown in figure at right. |
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| + | The inverse function [[ArcTetration]] is denoted with symbol ate; \( \mathrm{ate}_b = \mathrm{tet}_b^{-1} \).<br> |
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| − | Case of complex values of \(b\) is under investigation; conditions, that make the solution of equations above unique, may have need to be reformulated. |
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| + | The iterates of [[Exponential]] can be expressed as follows: |
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| + | \( |
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| + | \exp_b^{\ n}=\operatorname{tet}_b\big(n+\operatorname{ate}_b(z)\big) |
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| + | \). |
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| + | Here, number \(n\) of the iterate has no need to be integer. In particular, \(\varphi=\exp^{1/2}\) |
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| + | appears as solution of equation \( \varphi(\varphi(x))=\mathrm e^x \); this equation had been considered by [[Hellmuth Kneser]] <ref name="k"> |
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| + | http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi |
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| + | [[Hellmuth Kneser]]. Reelle analytische Lösungen der Gleichung \( \varphi(\varphi(x))=\mathrm e^x \) und verwandter Funktionalgleichungen. |
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| + | Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67 |
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| + | </ref>, 1950. |
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| + | ==Notations== |
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| − | ==Tetration as [[superfunction]]== |
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| − | The tetration is [[superfunction]] of the [[exponential]]. The tetration can be interpreted as iterated exponential applied to unity. In the language [[Mathematica]], there already exists the special notation for the superfunction, id est, the function '''Nest'''. In particular, |
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| − | \({\rm tet}(z)={\rm Nest}[\exp,1,z]\) . |
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| − | Up to year 2025, '''Nest''' is implemented only for the integer values of the last argument; the use with any expression, different from an integer constant, causes the error message. |
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| + | The name [[tetration]] reflects its role as the next operation after exponentiation within the [[hyperoperation]] hierarchy:<br> |
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| − | For the beginning of year 2011, the tetration has been analyzed mainly for the real values of base \(b \! > \! 1\). |
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| + | [[Ackermann]]\(_1\), id est, the [[Addition]] appears as [[superfunction]] of [[unity increment]];<br> |
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| − | The different algorithms are used for the evaluation at |
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| + | [[Ackermann]]\(_2\), id est, the [[Multiplication]] appears as [[superfunction]] of additon; <br> |
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| + | [[Ackermann]]\(_3\), id est, the [[Exponential]] \(\exp\) appears as [[superfunction]] of multiplication;<br> |
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| + | [[Ackermann]]\(_4\), id est, the [[Tetration]] \(\mathrm{tet}\) appears as [[superfunction]] of [[exponential]];<br> |
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| + | [[Ackermann]]\(_5\), id est, the [[Pentation]] \(\mathrm{pen}\) appears as [[superfunction]] of [[tetration]];<br> |
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| + | and so on. These functions are qualified as [[ackermann]]s after the last name of |
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| + | mathematician [[Wilhelm Ackermann]]. |
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| + | The most studied case is the [[Natural tetration]] |
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| − | \(1 \! < \! b \! < \! \exp(1/\rm e)\) |
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| − | <ref name=" |
+ | <ref name="analuxp"> |
| − | + | https://www.ams.org/journals/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br> |
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| − | http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf <br> |
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http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf |
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf |
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| + | Dmitrii Kouznetsov. Solution of F(z+1)=exp(F(z)) in complex z-plane. |
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| − | D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. |
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| + | Mathematics of Computation, 2009, V.78, p.1647-1670. |
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| − | </ref>, |
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| + | </ref><ref name="vladi"> |
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| + | https://mizugadro.mydns.jp/PAPERS/2010vladie.pdf<br> |
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| + | https://www.vmj.ru/articles/2010_2_4.pdf <br> |
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| + | https://mizugadro.mydns.jp/PAPERS/2010_2_4.pdf |
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| + | D.Kouznetsov. Tetration as special function. (In Rusian) [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45. |
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| + | </ref>. It corresponds to base \(b=\mathrm e\); it can be written simply as |
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| + | \[ |
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| + | \operatorname{tet}(z)=\operatorname{tet}_{\mathrm e}(z). |
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| + | \] |
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| + | For base \(\mathrm e\) the first five [[ackermann]]s are shown in figure at right.<br> |
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| + | The dashed line refers to the [[natural tetration]]. |
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| + | In such a way, all the [[ackermann]]s are numerated. <br> |
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| − | \(b \! = \! \exp(1/\rm e)\approx 1.44\) |
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| + | In notation [[Ackermann]]\(_m (z)\), the number \(m\) is supposed to be positive integer (natural number). <br> |
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| − | <ref name="e1e"> |
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| + | To year 2025, the generalization of [[Ackermann]]\(_m(z)\) for non-integer values of \(m\) is not yet developed. |
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| − | http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf <br> |
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| + | However, the argument \(z\) may have complex values. |
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| − | http://mizugadro.mydns.jp/PAPERS/2011e1e.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). [[Mathematics of computation]], 2012 February 8. ISSN 1088-6842(e) ISSN 0025-5718(p) |
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| − | </ref>, |
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| + | [[Complex map]]s of [[tetration]] to various bases are shown in figures below. |
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| − | and for |
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| + | Lines \(u=\mathrm{const} \) and |
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| − | \(b \! >\! \exp(1/\rm e)\) |
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| + | lines \(v=\mathrm{const} \) are drawn for \(u+\mathrm i v=\mathrm{tet}_b(x\!+\!\mathrm i y)\). |
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| − | <ref name="moc1"> |
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| − | http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html D.Kouznetsov. (2009). Solutions of \(F(z+1)=\exp(F(z))\) in the complex plane.. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. </ref><ref name="vladi"> |
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| − | http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf |
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| − | D.Kouznetsov. [[Superexponential]] as [[special function]]. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45. |
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| − | </ref><ref name="aeq"> |
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| − | http://www.springerlink.com/content/u7327836m2850246/ |
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| − | H.Trappmann, D.Kouznetsov. Uniqueness of Analytic [[Abel Function]]s in Absence of a Real Fixed Point. [[Aequationes Mathematicae]], v.81, p.65-76 (2011)</ref>. |
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| + | ==Definition== |
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| − | Therefore, one could expect some peculiarity (perhaps, a [[branch point]]) of \({\rm tet}_b(z)\) at |
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| − | \(b\!=\!\exp(1/\rm e)\). |
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| + | Let \(T_b(z)=b^z\). |
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| − | On the other hand, the plots of this function look smooth; one might expect that at a fixed value of \(z\), \({\rm tet}_b(z)\) is [[holomorphic]] (analytic) with respect to \(b\), and, in particular, in vicinity of \(b \! \approx \! \exp(1/\rm e)\). However, the visual smoothness at the real values of \(b\) can substitute neither the investigation of the analytic properties for complex \(b\) nor the mathematical proof of [[holomorphizm]]. |
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| + | A function \(F\) is a ''[[superexponential]]'' (a [[superfunction]] of \(T_b\)) if |
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| + | \[ |
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| + | F(z+1) = T_b(F(z)). |
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| + | \] |
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| + | A [[tetration]] to real base \(b\) is real-holomorphic superexponential \(F\) |
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| − | ==Behavior at \(1\!<\!b\!<\exp(1/{\rm e})\)== |
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| + | such that \(F(x+\mathrm i y)\) remains bounded at \(y\to\pm\infty\) and \(F(0)=1\). |
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| − | At \(1\!<\!b\!<\exp(1/{\rm e})\), tetration is [[superfunction]] constructed with [[regular iteration]] at smallest of the real [[fixed point]]s \(L\!=\!L_1\), \(L\!=\!L_2\) of \(\log_b\). |
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| − | In language [[Mathematica]], the value of the fixed point can be expressed with function ProductLog, |
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| − | : \(L_1(b)=-(\mathrm{ProductLog}[- \mathrm{Log}[b]]/\mathrm{Log}[b])\). |
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| + | Such a solution is believed to exists and to be unique <ref> |
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| − | The tetration is [[periodic function]]; the period |
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| + | https://link.springer.com/article/10.1007/s00010-010-0021-6 |
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| − | : \(T=\frac{2\pi \mathrm i} {\ln(L \, \ln(b))} \) |
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| + | https://mizugadro.mydins.jp/2011uniabel.pdf |
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| − | is pure imaginary. The cut line \(\{x \in \mathbb R: x\le -2\}\) is reproduced at the translations along the imaginary axis. |
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| + | H.Trappmann, D.Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair. |
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| + | [[Aequationes Mathematicae]], v.81, p.65-76 (2011) |
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| + | </ref><ref> |
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| + | https://math.stackexchange.com/questions/284868/uniqueness-of-tetration |
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| + | Let 𝑓(0)=1 |
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| + | and 𝑓(𝑥+1)=2^𝑓(𝑥) // |
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| + | Also let f be infinitely differentiable. Then does f exist and is it unique? // |
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| + | If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 satisfies the conditions(if f is defined in [0,1] ,we can use the property to define it everywhere else). Similar things can be said for differentiability. But I don't how to solve the problem if it's infinitely differentiable. |
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| + | </ref>. |
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| + | ==Special cases== |
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| − | \(\mathrm{tet}_b(x)\) remains real at \(x>-2\). It has logarithmic singularity at \(x=-2\), and it is [[monotonous function}monotonous]]ly grows at |
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| + | <div class="thumb tright" style="float:right; margin:-6px 0px 4px 8px; background-color:#fff; width:480px; line-height:8px;"> |
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| − | \(x>-2\). |
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| + | {{pic|E1efig09abc1a150.png|480px}}<small><center>[[Complex map]]s of [[tetration]] \(\mathrm{tet}_b\) to base \(b\!=\!1.5\) , left; \(b\!=\!\exp(1/\mathrm e)\) , center ; \(b\!=\!\sqrt{2}\) , right. [[Суперфункции]]<ref name="ru"/>,с.251,Рис.17.4; [[Superfunctions]]<ref name="en"/>,p.245,Fig.17.4; |
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| + | </center></small> |
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| + | </div> |
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| + | <div class="thumb tright" style="float:right; margin:4px 0px 8px 8px; width:480px;"> |
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| + | {{pic|Tetsheldonmap03.png|480px}}<small><center>[[Tetration to Sheldon base]]. |
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| + | [[Суперфункции]]<ref name="ru"/>,с.257,Рис.18.3; [[Superfunctions]]<ref name="en"/>,p.250, top of Fig.18.3.</center></small> |
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| + | </div> |
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| + | [[Decimal tetration]], \(b=10\). The routine for the evaluation is loaded as [[F4ten.cin]]. |
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| − | The tetration approaches its fixed point \(L_1=L_1(b)\) at large values of the real part of its argument (and, in particular, along the real axis). |
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| + | It is used to plot the top picture. |
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| + | [[Natural tetration]], \(b=\mathrm e\). The [[complex map]] for this case is shown in figure above. The original description <ref name="analuxp"/> and the fast C++ implementation <ref name="vladie"> |
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| − | ==Evaluation and behavior at \(b\!=\! \exp(1/{\rm e})\)== |
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| + | https://www.emis.ams.org/journals/VMJ/articles/2010_2_4.pdf <br> |
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| + | https://mizugadro.mydns.jp/PAPERS/2010_2_4.pdf<br> |
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| + | https://mizugadro.mydns.jp/PAPERS/2010vladie.pdf<br> |
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| + | Д.Кузнецов. Тетрация как специальная функция. Владикавказский математический журнал, 2010, т.12, вып. 2, стр.31-45.<br> |
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| + | D.Kouznetsov, Tetration as special function. Vladikavkaz Mathematical Journal, 2010, |
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| + | v.12, Issue 2, p.33-45. |
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| + | </ref> are published and mentioned in book [[Superfunctions]] <ref name="en"/> |
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| + | [[Binary tetration]], \(b=2\), see [[Base 2]] |
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| − | For base \(b\!=\!\exp(1/{\rm e})\! \approx\! 1.44\), the standard [[regular iteration]] should be modified. |
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| + | [[tetration to base 1.5]], see [[Base 1.5]] |
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| − | In partucular, the "new expansion" |
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| − | <ref name="e1e"/> is suggested: |
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| − | : \(\displaystyle |
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| − | F(z)=\mathrm e\cdot\left(1-\frac{2}{z}\left( |
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| − | 1+\sum_{m=1}^{M} \frac{P_{m}\big(-\ln(\pm z) \big)}{(3z)^m} |
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| − | +\mathcal{O}\!\left(\frac{|\ln(z)|^{M+1}}{z^{M+1}}\right) |
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| − | \right) \right) \) |
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| − | where |
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| − | : \(\displaystyle P_{m}(t)=\sum_{n=0}^{m} ~c_{n,m~}~ t^n\) |
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| − | The substitution of into equation \(~ F(z\!+\!1)\!=\!\exp(F(z)/\mathrm e)~\) |
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| − | and the asymptotic analysis with small parameter \(|1/z|\) determines |
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| − | the coefficients \(c\) in the polynomials above. In particular, |
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| − | : \( \begin{array}{ccc} |
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| − | P_{1}(t)&=&t \\ |
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| − | P_{2}(t)&=&t^{2}+t+1/2 \\ |
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| − | P_{3}(t)&=&t^{3}+\frac{ 5}{ 2}t^{2}+\frac{ 5}{2}t +\frac{ 7}{10} \\ |
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| − | P_{4}(t)&=&t^{4}+\frac{13}{ 3}t^{3}+\frac{ 45}{6}t^{2}+\frac{53}{10}t +\frac{ 67}{60} \\ |
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| − | P_{5}(t)&=&t^{5}+\frac{77}{12}t^{4}+\frac{101}{6}t^{3}+\frac{83}{ 4}t^{2}+\frac{653}{60}t+\frac{2701}{1680} |
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| − | \end{array} \) |
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| − | The evaluation with 9 polynomials \(P\) gives an approximation of \(\tilde F(z)\) with 15 decimal digits at \(\Re(z)\!>\!4\). |
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| − | For small values of \(z\), the iterations of formula \(F(z)=\ln(F(z\!+\!1))\!~ \mathrm e\) |
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| − | can be used. |
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| + | [[Crytical tetration]], \(b= \mathrm e^{1/\mathrm e}\), see [[Base e1e]] |
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| − | The tetration can be approximated as |
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| − | : \(\mathrm{tet}_{\exp(1/\mathrm{e})}(z)= F(x_0\!+\!z)\) |
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| − | where \(x_0\) is solution of equation \(F(x_0)=1\). |
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| − | With the [[complex double]] precision, the resulting approximation returns of order of 14 correct decimal digits in the whole complex plane |
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| − | (except the singularities). |
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| + | [[Tetration to base sqrt2]], \(b= \sqrt{2}\), see [[Base sqrt2]] |
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| − | The tetration is aperiodic, holomorphic in the range \(\mathbb C \backslash \{ x\in \mathbb R : x\!\le\! -2 \}\). |
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| − | A large values of the argument, it approaches the limiting value \(\mathrm e\), independently on the direction. |
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| − | In particular, along the real axis, the the function shows the monotonic growth from \(-\infty\) at \(-2\) to \(\mathrm e\) at infinity, passing through points (-1,0) and (0,1). |
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| + | [[Tetration to Sheldon base]] \(b=1.52598338517+0.0178411853321\ \mathrm i\). |
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| − | At any \(z\) from the range of definition, In vicinity of \(b\!\approx\!\exp(1/\mathrm e)\), tetration \(\mathrm{tet}_b(z)\) seems to be continuous function of \(b\), although very different agorithms are used for the evaluation in the three cases, |
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| + | The original algorithm <ref name="analuxp"/> allows the straight-forward generalization for the case of complex values of base \(b\). After the request by [[Sheldon Levenstain]], the complex map of this tetration had been generated; it is shown in figure at right. |
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| − | \(1\!<\!b\!<\!\exp(1/\mathrm e)\), |
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| − | \(b=\exp(1/\mathrm e)\), and |
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| − | \(b\!>\!\exp(1/\mathrm e)\). The figure below collects the [[complex map]]s of tetration to base |
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| − | \(b\!=\!1.5\) , left, |
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| − | \(b\!=\!\exp(1/\mathrm e)\) , center, and |
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| − | \(b\!=\!\sqrt{2}\) , right. \(f\!=\! \mathrm{tet}_b(x\!+\!\mathrm i y)\) is shown in the \(x,y\) plane with levels |
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| − | \(~p\!=\!\Re(f)\!=\! \mathrm{const}~\) and levels |
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| − | \(~q\!=\!\Im(f)\!=\! \mathrm{const}~\). The integer values correspond to the thick lines. |
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| + | ==Continuity at base b=exp(1/e)== |
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| − | <div style="margin:8px 0px 8px -8px"> |
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| + | <div class="thumb tright" style="float:right; margin:0 0 8px 8px; width:280px; line-height:10px"> |
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| − | {{pic|E1efig09abc1a150.png|720px}}<small><center>[[Complex map]]s of \(\mathrm{tet}_{b}\) for \(b\!=\! 1.5\), \(\exp(1/\mathrm e)\) , \(\sqrt{2}\); \(p\!+\!\mathrm i q=\mathrm{tet}_b(x\!+\!\mathrm i y)\) |
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| + | {{pic|BlackSheep.png|280px}} |
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| − | </center></small> |
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| + | <small><center>"half-sheep". [[Суперфункции]]<ref name="ru"> |
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| + | https://mizugadro.mydns.jp/BOOK/202.pdf |
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| + | Дмитрий Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014 |
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| + | </ref>,с.218,Рис.15.6; [[Superfunctions]] <ref name="en"/>,p.215,Fig.15.6</center></small> |
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</div> |
</div> |
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| + | This cartoon at right illustrates a philosophical point in tetration theory:<br> |
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| − | ==Behavior at \(b\!\approx\! \exp(1/{\rm e})\), \(z\!\approx\! 0\)== |
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| + | <b>"The only we may conclude, that in this county, there is at least one sheep, |
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| − | If the \({\mathrm{tet}}_b(z)\) is holomorphic in some vicinity of \(b\!\approx \! \exp(1/{\rm e})\), \(z\!\approx 0\), then it is expandable into the Taylor series |
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| + | and at least the right-hand side of this animal is black"</b>. <br> |
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| − | :\(\displaystyle |
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| + | The heuristic assumptions that appear “obvious’’ (e.g., that the right side of a sheep has the same color as its left side) appears without rigorous proof and, from point of a mathematician, may happen to be wrong. |
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| − | {\mathrm {tet}}_b(x)=1+\sum_{n=1}^{N-1} \! |
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| − | \left(\sum_{m=0}^{M-1} \! |
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| − | c_{m,n} \beta^m + |
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| − | O(\beta^M) |
||
| − | \right) x^n |
||
| − | +{\mathcal O}(x^N) |
||
| − | \) |
||
| − | where |
||
| − | \(\beta = b\!-\! \exp(1/\mathrm e)\); the coefficients \(c\) of this expansion can be estimated fitting the evaluations of tetration, using the algorithms reported for real base. This leads to the following preliminary estimates: |
||
| − | :\( |
||
| − | \begin{array}{ccrrrrrr} |
||
| − | m&c_{m,0} &c_{m,1} ~ ~ ~& c_{m,2} ~ ~ ~ & c_{m,3} ~ ~ ~ &c_{m,4}~ ~ ~ \\ |
||
| − | 0 & 1 & 0.61061 &-0.23171 & 0.09225 &-0.03757\\ |
||
| − | 1 & 0 & 0.69521 & 0.41315 &-0.16027& 0.07007 \\ |
||
| − | 2 & 0 &-0.57851 & 0.18323 & 0.49162&-0.15216 \\ |
||
| − | 3 & 0 & 0.64730 &-0.62933 &-0.51128& 0.51372\\ |
||
| − | 4 & 0 &-0.84098 & 1.23261 & 0.42470&-0.97551 \\ |
||
| − | 5 & 0 & 1.19090 &-2.12653 &-0.06895& 1.57684 |
||
| − | \end{array} |
||
| − | \) |
||
| − | However, the \(\beta \) appears as a branch point of tetration, being considered as finction of base \(b\). So, the precision of such evaluation is not so high (perhaps, only two or three decimal digits are significant in the estimates above); for the efficient evaluation of the coefficients of such an expansion, the tetration to the complex base should be analyzed. Such an analysis could be matter for the further research. Any independent confirmation (or correction) of the approximations of the coefficients above should be appreciated. |
||
| + | The similar heuristic assumption refers to the continuity of [[tetration]] \(\mathrm{tet}_b(z) \) being considered as function of base \(b\) at point \(b=\exp(1/\mathrm e)\). |
||
| − | ==Behavior at \(b\!>\!\exp(1/\mathrm e)\!\approx 1.44\)== |
||
| + | The maps above for \(b=1.5\), \(b=\exp(1/\mathrm e)\approx 1.44\) and \(b=\sqrt{2}\approx 1.41\) |
||
| − | Tetration is holomorphic in the whole complex plane except part of the real axis \(\{ x\in \mathbb R : x\le -2 \}\). |
||
| − | + | as well as the explicit plot at the top make an impression, that the [[tetration]] is continuous and perhaps holomorphic with respect to \(b\) at this point: the variation in the 3d significant figure causes small change of the view of the curves; at least for moderate valies of argument \(z\) |
|
| − | As the real part of the argument grows, the tetration shows complicated, [[fractal]]-like structures. In the [[complex map]], the lines of zero real part are reproduced at the unity translations along the real axis, but the new and new details appear at each step. |
||
| + | As in the case of the sheep in the cartoon, this assumptions about tetration |
||
| − | == Case \(b\!=\! \mathrm{e}\)== |
||
| + | is not obvious and require careful justification. |
||
| + | The prelimninary numerical analysis indicates, that |
||
| − | <div class="thumb tright" style="float:right; margin:-30px 0px 8px 12px; background-color:#fff"> |
||
| − | + | \(\mathrm{tet}_b(z)\) for \(\Re(b)<\exp(1/\mathrm e)\) |
|
| + | is not analytical extension of |
||
| − | </div> |
||
| + | \(tet_b(z)\) for \(\Re(b)>\exp(1/\mathrm e)\); |
||
| − | The case of [[natural tetration]] refers to base \(b\!=\!\mathrm e\! \approx\! 2.71~\); this case is considered in years 2010, 2011 |
||
| + | in the simplest way, the cut line along line \(\Re(b)=\exp(1/\mathrm e)\) |
||
| − | <ref name="moc2009"> |
||
| + | divides the complex plane to two almost independent parts, |
||
| − | http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670. |
||
| + | and only in point \(Re(b)=\exp(1/\mathrm e)\), these two tetrqations have the same limiting |
||
| − | </ref><ref name="vladi"/>. The fast complex(double) implementation in [[C++]] is suggested. |
||
| + | tetration. <br> |
||
| + | However, the cut at the complex \(b\)-plane has no need to follow the vertical line \(\Re(b)=\exp(1/\mathrm e)\); but this choice is simplest and has priority (following the last, 6th of the [[TORI axioms]].) |
||
| + | ==Applications== |
||
| − | As in the case of exponential, in the indication of the natural tetration, the subscript can be omitted: |
||
| + | The obvious application of the tetration may refer to representation of huge numbers.<br> |
||
| − | \(\exp_{\mathrm{e}}=\exp\) and |
||
| + | For example the [[Googolplex]] number <ref> |
||
| − | \(\mathrm{tet}_{\mathrm{e}}=\mathrm{tet}\). |
||
| + | https://en.wikipedia.org/wiki/Googolplex |
||
| + | A '''googolplex''' is the [[large number]] \(10^{10^{100}}\), that is, 10 raised to the power of a [[googol]]. If written out in ordinary [[decimal notation]], it would be 1 followed by a [[googol]] (10<sup>100</sup>) zeroes – a physically impossible number to write explicitly. |
||
| + | </ref> can be expressed as follows: |
||
| + | \[ |
||
| + | 10^{10^{100}}=\mathrm{tet}_{10}^3(2) |
||
| + | \] |
||
| + | However, up to year 2025, Editor have not yet found any [[scientific concept]] |
||
| + | that deals with such a huge number. |
||
| + | The more realistic application may refer to the approximation of processes that |
||
| − | The figure shows the [[complex map]] of natural tetration; the real and the imaginary parts of |
||
| + | grows faster than any polynomial but slower than any exponential. |
||
| − | \(f\!=\!\mathrm {tet}(x+\mathrm i y)\) are shown in the \(x\),\(y\) plane with levels |
||
| + | The non-integer iterates greatly extend the arsenal of functions available for |
||
| − | \(u\!=\! \Re(f)\!=\! \mathrm {const}\) and |
||
| + | construction of efficient fits, involving less fitting parameters and/or providing better precision and/or having wider range of approximation. |
||
| − | \(v\!=\! \Im(f)\!=\! \mathrm {const}\). |
||
| + | The \(n\)-th iterate of the exponential: |
||
| − | In the upper half-plane, the tetration approaches the limiting value, which is [[fixed point]] \(L\) of natural [[logarithm]]; \(L\) is solution of \(L\!=\!\ln(L)\), and \(L\!\approx\! 0.318132 + 1.33724 \mathrm i\). |
||
| + | \[ |
||
| + | \exp_b^{\ n}(z) = \operatorname{tet}_b\!\big(n + \operatorname{ate}_b(z)\big), |
||
| + | \] |
||
| + | for real \(n\). One example of such a function is mentioned in the Preamble. |
||
| + | An additional extension could be generalization of [[ackermann]]\(m\) |
||
| − | In the lower half-plane, the tetration approaches the conjugated value, id est, \(L^*\!\approx\! 0.318132 - 1.33724 \mathrm i\). |
||
| + | to non-integer values of \(_m\), looking for the real-holomorphic solition \(A\) of equations |
||
| + | \[ |
||
| + | A_m(z+1)=A_{m-1}(A_m(z)) \\ |
||
| + | A_2(z)= \mathrm e\ z \\ |
||
| + | Z_m(0)=1 \text{ for } m>2 |
||
| + | \] |
||
| + | treating \(A_m(z)\) as holomorphic function of two variables \(m\) and \(z\).<br> |
||
| + | Such a generalization can be matter for the future research. |
||
| + | ==Acknowledgement== |
||
| − | ==Tetration to complex base== |
||
| + | [[ChatGPT]] helped to improve this article. |
||
| − | The method with the Cauchi equation |
||
| − | <ref name="moc1"/> |
||
| − | can be used also for the complex values of the base. |
||
| − | The following properties should be assumed: |
||
| − | : \(\displaystyle \lim_{y\rightarrow +\infty}\) \(\mathrm{tet}_b(x+\mathrm i y)= \mathrm{Filog}(a)\) |
||
| − | : \(\displaystyle \lim_{y\rightarrow -\infty}\) \(\mathrm{tet}_b(x+\mathrm i y)= \mathrm{Filog}(a^*)^*\) |
||
| − | where \(a=\ln(b)\) and [[Filog]] is holomorphic function, \(t=\mathrm{filog}(a)\) is solution of equation |
||
| − | :\( \log_b(f)=f\) |
||
| + | ==References== |
||
| − | The [[Filog]] function can be expressed through the [[Tania function]] and through the [[WrightOmega]] function as follows: |
||
| + | {{ref}} |
||
| + | https://en.wikipedia.org/wiki/Ackermann_function |
||
| − | : \(\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\mathrm{i}\big)}{-z}\) |
||
| + | This function is defined from the [[recurrence relation]] \(\operatorname{A}(m+1, n+1) = \operatorname{A}(m, \operatorname{A}(m+1, n)) |
||
| + | \) with appropriate [[Base case (recursion)|base cases]]. |
||
| + | https://en.wikipedia.org/wiki/Tetration |
||
| − | ===Sheldon base=== |
||
| + | In [[mathematics]], '''tetration''' (or '''hyper-4''') is an [[operation (mathematics)|operation]] based on [[iterated]], or repeated, [[exponentiation]]. There is no standard [[mathematical notation|notation]] for tetration, though [[Knuth's up arrow notation]] <math>\uparrow \uparrow</math> and the left-exponent <math>{}^{x}b</math> are common. |
||
| + | https://en.citizendium.org/wiki/Tetration |
||
| − | <div style="margin:-4px 0px 18px -8px"> |
||
| + | '''Tetration''' is a rapidly growing [[mathematical function]], which was introduced in the 20th century and proposed for the representation of huge numbers in the Mathematics of Computation. <!--For positive integer values of its argument <math>x</math>, tetration <math>\mathrm{tet}_b(x)</math> on base <math>b</math> can be defined with: <math>{\ \mathrm{tet}_b(x) = \ \atop {\ }} {{\underbrace{b^{b^{\cdot^{\cdot^{b}}}}}} \atop {x}}~</math> |
||
| − | {{pic|Tetsheldonmap03.png|900px}} |
||
| + | !--> |
||
<small> |
<small> |
||
| + | https://tetrationforum.org/search.php?action=results&sid=094d9363dba94b4cfc99575883e4e5bb&sortby=&order=desc |
||
| − | \( \phantom{1234567890} f=\mathrm{tet}_s(x\!+\!\mathrm i y)\) with |
||
| + | </small> [[Tetration Forum]] |
||
| − | lines \(u\!=\!\Re(f)\!=\!\mathrm{const}\) and |
||
| − | lines \(v\!=\!\Im(f)\!=\!\mathrm{const}\) for \( s=1.52598338517 + 0.0178411853321~{i}\) |
||
| − | </small> |
||
| − | </div> |
||
| + | https://www.numdam.org/item?id=BSMF_1919__47__161_0 |
||
| − | The [[complex map]] of |
||
| + | P. Fatou. Sur les ´equations fonctionnelles. Bulletin de la Soci´et´e |
||
| − | \(\mathrm{tet}_s\) is shown at figure above, where \(s\) is the [[Sheldon base]], |
||
| + | Math´ematique de France, 47 (1919), p. 161-271. |
||
| − | \( s=1.52598338517 + 0.0178411853321~{i}\) . |
||
| + | https://eretrandre.org/rb/files/Ackermann1928_126.pdf |
||
| − | See the special article [[Tetration to Sheldon base]] for this case. |
||
| + | Wilhelm Ackermann. Zum Hilbertschen Aufbau der reellen Zahlen. [[Mathematische Annalen]] 99, Number 1(1928), Z.118-133. |
||
| + | <small>https://projecteuclid.org/journals/acta-mathematica/volume-100/issue-3-4/Regular-iteration-of-real-and-complex-functions/10.1007/BF02559539.full |
||
| − | The colored plots of various superexponentials for various complex values of base are available at the [[Tetration Fortum]] |
||
| + | </small> |
||
| − | by [[Sheldon Levenson]] |
||
| + | G.Szekeres. Regular iteration of real and complex functions. Acta Mathematica 1958, Volume 100, Issue 3-4, pp 203-258. |
||
| − | <ref name="sheldon"> |
||
| − | http://math.eretrandre.org/tetrationforum/showthread.php?tid=729 |
||
| − | Sheldon Levenson. Complex base tetration program. [[Tetration and Related Topics]], 2012 March 1. |
||
| − | </ref>; the question wether some of them are tetrations is in discussion. |
||
| + | https://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf |
||
| − | ==Fast growth== |
||
| + | W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188. |
||
| − | Along the real axis, the tetration to base \(b>\exp(1/\mathrm e)\) shows fast growth; it grows faster than any exponential. Due to such a growth, the tetration could be useful for the [[numerical representation of huge numbers]], that can be stored as \(\mathrm {tet}_b(x)\) with moderated values \(x\), keeping the stored value of the number distinguishable from \(+\infty\). The arithmetics for such "tetrational" variables should be implemented using the properties of the tetration, avoiding the conversion to the conventional floating-point form. |
||
| + | https://www.tandfonline.com/doi/full/10.1080/10652460500422247 |
||
| − | Perhaps, at least in century 21, the tetration satisfies the requirements of the [[computational mathematics]] in the rapidly growing functions; if not, the superfunction of tetration, id est, [[pentation]] (\(\mathrm {pen}_b\)) or even higher [[Ackermann function]]s are available. For natural base (\(b\!=\!\mathrm e\!=\!\exp(1) \approx 2.71\)), the [[pentation]] \(\mathrm{pen}_{\mathrm e}\) can be expressed through the [[regular iteration]], allowing the efficient evaluation while the tetration is already implemented <ref name="vladi"/>. |
||
| + | M.H.Hooshmand, (2006). Ultra power and ultra exponential functions. Integral [[Transforms and Special Functions]] 17 (8): 549–558 |
||
| − | <!-- |
||
| − | For the case of natural tetration, the complex double implementations in C++ and than in Mathematica are available at Citizendium; the generators of the figures posted include the codes of the routine used. The details of the implementation are described in the [[Vladikavkaz Mathematical Journal]] |
||
| − | !--> |
||
| + | https://sciencepublishinggroup.com/article/10.11648/j.acm.20140306.14 |
||
| − | ==Arctetration== |
||
| + | Dmitrii Kouznetsov. |
||
| − | [[Arctetration]] (ate) is inverse function of tetration; it is [[Abel function]] of the [[exponential]]. |
||
| + | Evaluation of Holomorphic Ackermanns. |
||
| − | Afctetration \(\mathrm {ate}_b\) to base \(b\) satisfies the relations |
||
| + | Published in [[Applied and Computational Mathematics]] (Volume 3, Issue 6) |
||
| − | : \(\mathrm{ate}_b(\mathrm{tet}_b(z))=z\) |
||
| + | Received: 21 November 2014 Accepted: 17 December 2014 Published: 27 December 2014, Page(s) 307-314. |
||
| − | : \(\mathrm{tet}_b(\mathrm{ate}_b(z))=z\) |
||
| − | at least in some ranges of values of \(z\). Also, the arctetration satisfies the [[Abel equation]] |
||
| − | : \(\mathrm{ate}_b(b^z)=\mathrm{ate}_b(z) +1\) |
||
| − | Roughly, the arctetration counts, how many times the logarithm should be taken of a value before the value becomes unity. |
||
| + | https://www.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pq-origsite=gscholar&cbl=18750 |
||
| − | While the tetration is [[superfunction]] of the [[exponential]], the arctetration is the [[Abel function]]. |
||
| + | [[Samuel Cowgill]]. EXPLORING TETRATION IN THE COMPLEX PLANE |
||
| + | EXPLORING TETRATION IN THE COMPLEX PLANE |
||
| + | A Thesis presented to the faculty of Arkansas State University in partial. |
||
| + | fulfillment of the requirements for the Degree of |
||
| + | MASTER OF SCIENCE IN MATHEMATICS |
||
| + | ARKANSAS STATE UNIVERSITY //MAY 2017 |
||
| + | Approved by: |
||
| + | Dr. [[William Paulsen]], Thesis Advisor |
||
| + | Dr. [[Jie Miao]], Committee Member |
||
| + | Dr. [[Jeongho Ahn]], Committee MemberPREVIE |
||
| − | ==Iterated exponential== |
||
| − | The [[superfunction]] and the [[Abel function]] of the exponential are established; they are tetration "tet" and its inverse function "ate". |
||
| − | Then, the iterated exponential can be expressed as follows: |
||
| − | : \( \exp_b^c(z)=\mathrm{tet}_b(n+ \mathrm{ate}_b(z))\) |
||
| − | In this expression, the number \(n\) of iterations has no need to be [[integer number|integer]]; however, at integer values, |
||
| − | : \(\exp_b^{-2}(z)=\log_b(\log_b(z))\) |
||
| − | : \(\exp_b^{-1}(z)=\log_b(z)\) |
||
| − | : \(\exp_b^{0}(z)= z\) |
||
| − | : \(\exp_b^{1}(z)=\exp_b(z)\) |
||
| − | : \(\exp_b^{2}(z)=\exp_b(b^z)\) , |
||
| − | at least for values of \(z\) in some range in vicinity of the real axis; the arctetration unavoidably has the cut lines in the complex plane. |
||
| − | |||
| − | <!--The representation through the tetration and arctetration allows to evaluate not only integer iterates but even the complex iterates of the exponential.!--> |
||
| − | The realizations of the super-exponentials and the Abel–exponential by [[Hellmuth Kneser]] |
||
| − | <ref name="kneser"> |
||
| − | <!--http://www.ils.uec.ac.jp/~dima/Relle.pdf!--> |
||
| − | http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 |
||
| − | H.Kneser. Reelle analytische Lösungen der Gleichung \(\varphi(\varphi(x))=e^x\). |
||
| − | Equationes Mathematicae, Journal fur die reine und angewandte Mathematik {\bf 187} 56–67 (1950)</ref> |
||
| − | also allow to express the non-integer iterate of the exponential, but the functions he had constructed are not real along the real axis. For the applications to the real numbers, at least for base \(b\!>\!1\), the representation of iterated exponential through the tetration and arctetration looks more convenient. |
||
| − | |||
| − | ==Representation of huge numbers== |
||
| − | The tetration and the iterated exponential can be used for the representation of huge numbers. |
||
| − | In partucular the number [[googol]] <ref name="googol"> |
||
| − | http://mathworld.wolfram.com/Googol.html Googol |
||
| − | </ref> which is \(10^{100}\) appears to be \(\exp_{10}^2(2) = 10^{10^2}\), while the number [[googolplex]] |
||
| − | <ref name="googolplex"> |
||
| − | http://mathworld.wolfram.com/Googolplex.html Googolplex |
||
| − | </ref> |
||
| − | appears as \(\exp_{10}^3(2)=10^{10^{10^2}}\) |
||
| − | |||
| − | |||
| − | ==[[Superfunctions]]== |
||
| − | Properties of tertration are described in book [[Superfunctions]] (2014–2020) |
||
| − | <ref> |
||
| − | http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf (a little bit out of date) <br> |
||
| − | http://mizugadro.mydns.jp/BOOK/468.pdf |
||
| − | D.Kouznetov. Superfunctions. Lambert Academic Publishing, 2020. |
||
| − | </ref> |
||
| − | |||
| − | After the appearance of the first version of Book [[Superfunctions]], certain advances are observed about evaluation of [[tetration]] |
||
| − | of complex argument; the new algorithm is suggested, that seems to be mode efficient, than the [[Cauchi integral]] described in the Book. |
||
| − | <ref> |
||
| − | http://journal.kkms.org/index.php/kjm/article/view/428 |
||
| − | William Paulsen. |
||
| − | Finding the natural solution to f(f(x))=exp(x). |
||
| − | Korean J. Math. Vol 24, No 1 (2016) pp.81-106. |
||
| − | </ref><ref> |
||
| − | https://link.springer.com/article/10.1007/s10444-017-9524-1 |
||
| − | William Paulsen, Samuel Cowgill. |
||
| − | Solving F(z + 1) = b F(z) in the complex plane. |
||
| − | Advances in Computational Mathematics, December 2017, Volume 43, Issue 6, pp 1261–1282 |
||
| − | </ref><ref> |
||
| − | https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pq-origsite=gscholar&cbl=18750&diss=y |
||
| − | Cowgill, Samuel. Exploring Tetration in the Complex Plane. |
||
| − | Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680. |
||
| − | </ref><ref> |
||
https://link.springer.com/article/10.1007/s10444-018-9615-7 |
https://link.springer.com/article/10.1007/s10444-018-9615-7 |
||
| − | William Paulsen. |
+ | William Paulsen. Tetration for complex bases. |
| + | Advances in Computational Mathematics. |
||
| − | Tetration for complex bases. |
||
| + | Published: 02 June 2018// |
||
| − | Advances in Computational Mathematics, 2018.06.02. |
||
| + | Volume 45, pages 243–267, (2019) |
||
| − | </ref> |
||
| − | |||
| − | ==Humor about tetration== |
||
| − | <div class="thumb tleft" style="float:left; margin:-4px 8px 0px -8px; width:300px"> |
||
| − | {{pic|BlackSheep.png|300px}}<small><center>.. Your generalisation seems to me not supported. All, that we can deduce from this observation, is, that in this country, there exist at least one sheep; and at least the right-hand side of this sheep is black.</center></small> |
||
| − | </div> |
||
| − | |||
| − | {{pic|2x2uni100.jpeg|260px}} |
||
| − | |||
| − | ==References== |
||
| − | {{ref}} |
||
| − | |||
| − | http://www.proofwiki.org/wiki/Definition:Tetration <br> |
||
| − | http://en.citizendium.org/wiki/Tetrational <br> |
||
| − | http://en.wikipedia.org/wiki/Tetration<br> |
||
| − | http://math.eretrandre.org/tetrationforum/index.php <br> |
||
| − | http://samlib.ru/k/kuznecow_d_j/ackermann.shtml<br> |
||
| − | http://www.tetration.org/Ackermann/arithmetic.html [[Daniel Geisler]]. All is Arithmetic. (2009) |
||
| − | |||
| − | https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br><!-- no more valid |
||
| − | http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>!--> |
||
| − | http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014. In Russian.<br> |
||
| − | http://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020. |
||
| − | |||
| − | http://myweb.astate.edu/wpaulsen/tetration2.pdf <br> |
||
| − | http://link.springer.com/article/10.1007/s10444-017-9524-1 <br> |
||
| − | [[William Paulsen]] and [[Samuel Cowgill]]. |
||
| − | Solving \(F(z+1)=b^{F (z)}\) in the complex plane. |
||
| − | [[Advances in Computational Mathematics]], 2017 March 7, p. 1–22 |
||
| − | |||
| − | 2018. https://doi.org/10.1080/10236198.2017.1307350 M.H. Hooshmand. (2018) Ultra power of higher orders and ultra exponential functional sequences. [[Journal of Difference Equations and Applications]]. Volume 24, 2018 - Issue 5: Special Issue: [[European Conference on Iteration Theory]] 2016. Special Issue Editors: Marek Cezary Zdun & Jaroslav Smital. |
||
| + | https://www.researchgate.net/publication/325532999_Tetration_for_complex_bases |
||
| − | 2019. https://doi.org/10.1007/s10444-018-9615-7 W.Paulsen. Tetration for complex bases. Adv Comput Math 45, 243–267 (2019). |
||
| + | <br> |
||
| + | https://www.researchgate.net/figure/Level-curves-for-Rk-b-z-and-Ik-b-z-0-1-2-3-4-for-b-i_fig1_325532999 |
||
| + | [[William Harold Paulsen]]. Tetration for complex bases. |
||
| + | Advances in Computational Mathematics 45(6), |
||
| + | February 2019 |
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| + | DOI:10.1007/s10444-018-9615-7 |
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| + | http://myweb.astate.edu/wpaulsen/tetration2.pdf |
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| − | 2021. https://arxiv.org/abs/2101.03021v2 [[James David Nixon]]. Hyper-operations By Unconventional Means. March 8, 2021 |
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| + | [[William Paulsen]] and [[Samuel Cowgil]]. |
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| + | Solving F(z+1)=b^F(z) in the complex plane. |
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{{fer}} |
{{fer}} |
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==Keywords== |
==Keywords== |
||
| − | «[[Abel function]]», |
+ | «[[Abel function]]», |
| − | «[[Ackermann |
+ | «[[Ackermann]]», |
| − | «[[ |
+ | «[[ate]]», |
| − | «[[ |
+ | «[[Base e1e]]», |
| − | «[[Base |
+ | «[[Base sqrt2]]», |
| − | «[[ |
+ | «[[Exponential]]», |
| − | «[[ |
+ | «[[Filog]]», |
| − | «[[ |
+ | «[[fit1.cin]]», |
| − | «[[ |
+ | «[[Fixed point]]», |
| − | «[[ |
+ | «[[Fsexp.cin]]», |
| − | «[[ |
+ | «[[Fslog.cin]]», |
| − | «[[ |
+ | «[[Hellmuth Kneser]]», |
| − | «[[ |
+ | «[[Iterates]]», |
| − | «[[ |
+ | «[[Iterate of exponential]]», |
| − | «[[ |
+ | «[[Maps of tetration]]», |
| − | «[[ |
+ | «[[Mathematics of Computation]]», |
| − | «[[ |
+ | «[[Logarithm]]», |
| + | «[[Superfunction]]», |
||
| − | «[[Natural tetration]]» (to base \(b\!=\!\mathrm e\!\approx \! 2.71\)), |
||
| − | «[[ |
+ | «[[Superfunctions]]», |
| − | «[[ |
+ | «[[tet]]», |
| − | «[[ |
+ | «[[Tetration]]», |
| − | «[[ |
+ | «[[Transfer equation]]», |
| − | «[[Theorem on asymptotic increment]]», |
||
| − | «[[Transfer equation]]», |
||
| − | «[[WrightOmega]]», |
||
| − | «[[ate]]», |
||
| − | «[[tet]]», |
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| − | |||
| − | «[[Суперфункции]]», |
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| − | |||
| − | [[Category:Ackermann function]] |
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| − | [[Category:Book]] |
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[[Category:English]] |
[[Category:English]] |
||
| − | [[Category: |
+ | [[Category:Exponential]] |
| − | [[Category: |
+ | [[Category:Superexponential]] |
| − | [[Category:Pentation]] |
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| − | [[Category:Science]] |
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| − | [[Category:Tetration]] |
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| − | [[Category:Special function]] |
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[[Category:Superfunction]] |
[[Category:Superfunction]] |
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[[Category:Superfunctions]] |
[[Category:Superfunctions]] |
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| + | [[Category:Special function]] |
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| + | [[Category:Tetration]] |
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Revision as of 15:31, 13 December 2025
Tetration (Тетрация) is the superfunction of the exponential map. For a given base \(b\), the tetration \(\operatorname{tet}_b\) is defined as the function satisfying the transfer equation \[ \operatorname{tet}_b(z+1) = b^{\operatorname{tet}_b(z)} \] together with additional normalization and regularity conditions that select a unique solution among infinitely many possible superfunctions. For real values of the argunet, the explicit plots of \(\operatorname{tet}_b(x)\) versos \(x\) for various real values of base \(b\) is shown in figure at right.
The inverse function ArcTetration is denoted with symbol ate; \( \mathrm{ate}_b = \mathrm{tet}_b^{-1} \).
The iterates of Exponential can be expressed as follows:
\(
\exp_b^{\ n}=\operatorname{tet}_b\big(n+\operatorname{ate}_b(z)\big)
\).
Here, number \(n\) of the iterate has no need to be integer. In particular, \(\varphi=\exp^{1/2}\)
appears as solution of equation \( \varphi(\varphi(x))=\mathrm e^x \); this equation had been considered by Hellmuth Kneser [3], 1950.
Notations
The name tetration reflects its role as the next operation after exponentiation within the hyperoperation hierarchy:
Ackermann\(_1\), id est, the Addition appears as superfunction of unity increment;
Ackermann\(_2\), id est, the Multiplication appears as superfunction of additon;
Ackermann\(_3\), id est, the Exponential \(\exp\) appears as superfunction of multiplication;
Ackermann\(_4\), id est, the Tetration \(\mathrm{tet}\) appears as superfunction of exponential;
Ackermann\(_5\), id est, the Pentation \(\mathrm{pen}\) appears as superfunction of tetration;
and so on. These functions are qualified as ackermanns after the last name of
mathematician Wilhelm Ackermann.
The most studied case is the Natural tetration
[4][5]. It corresponds to base \(b=\mathrm e\); it can be written simply as
\[
\operatorname{tet}(z)=\operatorname{tet}_{\mathrm e}(z).
\]
For base \(\mathrm e\) the first five ackermanns are shown in figure at right.
The dashed line refers to the natural tetration.
In such a way, all the ackermanns are numerated.
In notation Ackermann\(_m (z)\), the number \(m\) is supposed to be positive integer (natural number).
To year 2025, the generalization of Ackermann\(_m(z)\) for non-integer values of \(m\) is not yet developed.
However, the argument \(z\) may have complex values.
Complex maps of tetration to various bases are shown in figures below. Lines \(u=\mathrm{const} \) and lines \(v=\mathrm{const} \) are drawn for \(u+\mathrm i v=\mathrm{tet}_b(x\!+\!\mathrm i y)\).
Definition
Let \(T_b(z)=b^z\). A function \(F\) is a superexponential (a superfunction of \(T_b\)) if \[ F(z+1) = T_b(F(z)). \]
A tetration to real base \(b\) is real-holomorphic superexponential \(F\) such that \(F(x+\mathrm i y)\) remains bounded at \(y\to\pm\infty\) and \(F(0)=1\).
Such a solution is believed to exists and to be unique [6][7].
Special cases
Decimal tetration, \(b=10\). The routine for the evaluation is loaded as F4ten.cin. It is used to plot the top picture.
Natural tetration, \(b=\mathrm e\). The complex map for this case is shown in figure above. The original description [4] and the fast C++ implementation [8] are published and mentioned in book Superfunctions [2]
Binary tetration, \(b=2\), see Base 2
tetration to base 1.5, see Base 1.5
Crytical tetration, \(b= \mathrm e^{1/\mathrm e}\), see Base e1e
Tetration to base sqrt2, \(b= \sqrt{2}\), see Base sqrt2
Tetration to Sheldon base \(b=1.52598338517+0.0178411853321\ \mathrm i\). The original algorithm [4] allows the straight-forward generalization for the case of complex values of base \(b\). After the request by Sheldon Levenstain, the complex map of this tetration had been generated; it is shown in figure at right.
Continuity at base b=exp(1/e)
This cartoon at right illustrates a philosophical point in tetration theory:
"The only we may conclude, that in this county, there is at least one sheep,
and at least the right-hand side of this animal is black".
The heuristic assumptions that appear “obvious’’ (e.g., that the right side of a sheep has the same color as its left side) appears without rigorous proof and, from point of a mathematician, may happen to be wrong.
The similar heuristic assumption refers to the continuity of tetration \(\mathrm{tet}_b(z) \) being considered as function of base \(b\) at point \(b=\exp(1/\mathrm e)\).
The maps above for \(b=1.5\), \(b=\exp(1/\mathrm e)\approx 1.44\) and \(b=\sqrt{2}\approx 1.41\) as well as the explicit plot at the top make an impression, that the tetration is continuous and perhaps holomorphic with respect to \(b\) at this point: the variation in the 3d significant figure causes small change of the view of the curves; at least for moderate valies of argument \(z\)
As in the case of the sheep in the cartoon, this assumptions about tetration is not obvious and require careful justification.
The prelimninary numerical analysis indicates, that
\(\mathrm{tet}_b(z)\) for \(\Re(b)<\exp(1/\mathrm e)\)
is not analytical extension of
\(tet_b(z)\) for \(\Re(b)>\exp(1/\mathrm e)\);
in the simplest way, the cut line along line \(\Re(b)=\exp(1/\mathrm e)\)
divides the complex plane to two almost independent parts,
and only in point \(Re(b)=\exp(1/\mathrm e)\), these two tetrqations have the same limiting
tetration.
However, the cut at the complex \(b\)-plane has no need to follow the vertical line \(\Re(b)=\exp(1/\mathrm e)\); but this choice is simplest and has priority (following the last, 6th of the TORI axioms.)
Applications
The obvious application of the tetration may refer to representation of huge numbers.
For example the Googolplex number [9] can be expressed as follows:
\[
10^{10^{100}}=\mathrm{tet}_{10}^3(2)
\]
However, up to year 2025, Editor have not yet found any scientific concept
that deals with such a huge number.
The more realistic application may refer to the approximation of processes that grows faster than any polynomial but slower than any exponential. The non-integer iterates greatly extend the arsenal of functions available for construction of efficient fits, involving less fitting parameters and/or providing better precision and/or having wider range of approximation.
The \(n\)-th iterate of the exponential: \[ \exp_b^{\ n}(z) = \operatorname{tet}_b\!\big(n + \operatorname{ate}_b(z)\big), \] for real \(n\). One example of such a function is mentioned in the Preamble.
An additional extension could be generalization of ackermann\(m\)
to non-integer values of \(_m\), looking for the real-holomorphic solition \(A\) of equations
\[
A_m(z+1)=A_{m-1}(A_m(z)) \\
A_2(z)= \mathrm e\ z \\
Z_m(0)=1 \text{ for } m>2
\]
treating \(A_m(z)\) as holomorphic function of two variables \(m\) and \(z\).
Such a generalization can be matter for the future research.
Acknowledgement
ChatGPT helped to improve this article.
References
- ↑ 1.0 1.1 1.2 1.3 https://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2014
- ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862
https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020. - ↑ http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi Hellmuth Kneser. Reelle analytische Lösungen der Gleichung \( \varphi(\varphi(x))=\mathrm e^x \) und verwandter Funktionalgleichungen. Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67
- ↑ 4.0 4.1 4.2
https://www.ams.org/journals/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf Dmitrii Kouznetsov. Solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, 2009, V.78, p.1647-1670. - ↑
https://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
https://www.vmj.ru/articles/2010_2_4.pdf
https://mizugadro.mydns.jp/PAPERS/2010_2_4.pdf D.Kouznetsov. Tetration as special function. (In Rusian) Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. - ↑ https://link.springer.com/article/10.1007/s00010-010-0021-6 https://mizugadro.mydins.jp/2011uniabel.pdf H.Trappmann, D.Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair. Aequationes Mathematicae, v.81, p.65-76 (2011)
- ↑ https://math.stackexchange.com/questions/284868/uniqueness-of-tetration Let 𝑓(0)=1 and 𝑓(𝑥+1)=2^𝑓(𝑥) // Also let f be infinitely differentiable. Then does f exist and is it unique? // If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 satisfies the conditions(if f is defined in [0,1] ,we can use the property to define it everywhere else). Similar things can be said for differentiability. But I don't how to solve the problem if it's infinitely differentiable.
- ↑
https://www.emis.ams.org/journals/VMJ/articles/2010_2_4.pdf
https://mizugadro.mydns.jp/PAPERS/2010_2_4.pdf
https://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
Д.Кузнецов. Тетрация как специальная функция. Владикавказский математический журнал, 2010, т.12, вып. 2, стр.31-45.
D.Kouznetsov, Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, Issue 2, p.33-45. - ↑ https://en.wikipedia.org/wiki/Googolplex A googolplex is the large number \(10^{10^{100}}\), that is, 10 raised to the power of a googol. If written out in ordinary decimal notation, it would be 1 followed by a googol (10100) zeroes – a physically impossible number to write explicitly.
https://en.wikipedia.org/wiki/Ackermann_function This function is defined from the recurrence relation \(\operatorname{A}(m+1, n+1) = \operatorname{A}(m, \operatorname{A}(m+1, n)) \) with appropriate base cases.
https://en.wikipedia.org/wiki/Tetration In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation \(\uparrow \uparrow\) and the left-exponent \({}^{x}b\) are common.
https://en.citizendium.org/wiki/Tetration Tetration is a rapidly growing mathematical function, which was introduced in the 20th century and proposed for the representation of huge numbers in the Mathematics of Computation.
https://tetrationforum.org/search.php?action=results&sid=094d9363dba94b4cfc99575883e4e5bb&sortby=&order=desc Tetration Forum
https://www.numdam.org/item?id=BSMF_1919__47__161_0 P. Fatou. Sur les ´equations fonctionnelles. Bulletin de la Soci´et´e Math´ematique de France, 47 (1919), p. 161-271.
https://eretrandre.org/rb/files/Ackermann1928_126.pdf Wilhelm Ackermann. Zum Hilbertschen Aufbau der reellen Zahlen. Mathematische Annalen 99, Number 1(1928), Z.118-133.
https://projecteuclid.org/journals/acta-mathematica/volume-100/issue-3-4/Regular-iteration-of-real-and-complex-functions/10.1007/BF02559539.full G.Szekeres. Regular iteration of real and complex functions. Acta Mathematica 1958, Volume 100, Issue 3-4, pp 203-258.
https://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188.
https://www.tandfonline.com/doi/full/10.1080/10652460500422247 M.H.Hooshmand, (2006). Ultra power and ultra exponential functions. Integral Transforms and Special Functions 17 (8): 549–558
https://sciencepublishinggroup.com/article/10.11648/j.acm.20140306.14 Dmitrii Kouznetsov. Evaluation of Holomorphic Ackermanns. Published in Applied and Computational Mathematics (Volume 3, Issue 6) Received: 21 November 2014 Accepted: 17 December 2014 Published: 27 December 2014, Page(s) 307-314.
https://www.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pq-origsite=gscholar&cbl=18750 Samuel Cowgill. EXPLORING TETRATION IN THE COMPLEX PLANE EXPLORING TETRATION IN THE COMPLEX PLANE A Thesis presented to the faculty of Arkansas State University in partial. fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN MATHEMATICS ARKANSAS STATE UNIVERSITY //MAY 2017 Approved by: Dr. William Paulsen, Thesis Advisor Dr. Jie Miao, Committee Member Dr. Jeongho Ahn, Committee MemberPREVIE
https://link.springer.com/article/10.1007/s10444-018-9615-7 William Paulsen. Tetration for complex bases. Advances in Computational Mathematics. Published: 02 June 2018// Volume 45, pages 243–267, (2019)
https://www.researchgate.net/publication/325532999_Tetration_for_complex_bases
https://www.researchgate.net/figure/Level-curves-for-Rk-b-z-and-Ik-b-z-0-1-2-3-4-for-b-i_fig1_325532999
William Harold Paulsen. Tetration for complex bases.
Advances in Computational Mathematics 45(6),
February 2019
DOI:10.1007/s10444-018-9615-7
http://myweb.astate.edu/wpaulsen/tetration2.pdf William Paulsen and Samuel Cowgil. Solving F(z+1)=b^F(z) in the complex plane.
Keywords
«Abel function», «Ackermann», «ate», «Base e1e», «Base sqrt2», «Exponential», «Filog», «fit1.cin», «Fixed point», «Fsexp.cin», «Fslog.cin», «Hellmuth Kneser», «Iterates», «Iterate of exponential», «Maps of tetration», «Mathematics of Computation», «Logarithm», «Superfunction», «Superfunctions», «tet», «Tetration», «Transfer equation»,