imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = plot of the [[natural pension]] <math>\mathrm{pen}=\mathrm{pen}_{\mathrm e}</math>, id set, [[pentation]] to base <maht>\mathrm e=\exp(1)\approx 2.71</math>, id set, [[pentation]] to base <math>\mathrm e=\exp(1)\approx 2.71</math>; the thik black curve shows <math>y=\mathrm{pen}(x)</math>; the thik black curve shows <math>y=\mathrm{pen}(x)</math>.
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| The thin curves show the two asymptotics of pentation and the error <math>delta</math> of the linear approximation <math> z \mapsto 1+z</math>
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2014
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| |pub-country = Japan, Germany
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| |notes = [[Pentation]] is described at [[TORI]], http://mizugadro.mydns.jp/t/index.php/Pentation and also (In Russian) in the book <ref>
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| https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
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| http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
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| http://mizugadro.mydns.jp/BOOK/202.pdf
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| Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014. (In Russian), page 268, Figure 19.3.
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| </ref>
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| |versions = The image is borrowed from [[TORI]], http://mizugadro.mydns.jp/t/index.php/File:Penplot.jpg
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| }}
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| == Licensing ==
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| {{CC|by|3.0}}
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| ==Description==
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| [[Pentation]] pen is [[superfunction]] of [[tetration]] to the same base. Natural pentation is solution $F$ of the [[transfer equation]]
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| <math>
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| F(z\!+\!1)=\mathrm{tet}\Big( F(z))
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| </math>
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| constructed with [[regular iteration]] at the smallest real [[fixed point]] <math>L</math> of [[tetration]]; <math>L\approx -1.8503545290271812</math> is solution of equation
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| <math>L=\mathrm{tet}(L)</math>
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| with additional condition <math>F(0)=1</math>.
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| The real-real plot <math>y=\mathrm {pen}(x)</math> is shown with thick black curve.
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| The thin curves show approximations of pentation.
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| The red horizontal line shows the fixed point of tetration, <math>y=L</math>.
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| The thin blue curve shows the asymptotic of pentation at large negative values of the real part of the argument,
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| <math>
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| y= L+\exp(k(x+x_1))
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| </math>
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| where <math>k\approx 1.86573322821</math>
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| and <math>x_1 \approx 2.24817451898</math>
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| The thin green line shown the deviation from the linear approximation
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| <math>\mathrm{linear}(x)=1+x</math>
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| The deviation is denoted as <math>~\delta(x)=\mathrm{pen}(x)-\mathrm{linear}(x)</math>
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| In the range <math>-2.1\!<\!x\!<\!1.1</math>, the deviation is small, the linear approximation provides 2 correct significant digits. In order to make the deviation visible, it is scaled with factor 10, so, <math>y=10\delta(x)</math> is plotted.
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| Properties of tetration are described in publications
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| <ref>
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| http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
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| http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
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| http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.
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| </ref>
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| The [[regular iteration]] in construction of [[superfunction]] is described at [[TORI]], http://mizugadro.mydns.jp/t/index.php/Regular_iteration
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| and also in
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| <ref>
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| http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html 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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| </ref><ref>
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| http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
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| </ref><ref>
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| http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
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| http://tori.ils.uec.ac.jp/2012OR/2012or.pdf D. Kouznetsov. Superfunctions for optical amplifiers. Preprint ILS UEC, 2012
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| </ref>.
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| ==References==
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| <references/>
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