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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| |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 = The thin curves show the two asymptotics of pentation and the error of the linear approximation
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| |versions = 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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