imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Explicit plot]] of <math>n</math>th Iteration of the quadratic function, id est, [[power function]] for power 2; <math>y=\mathrm{Pow}_2^{~n}(z)=T^n(x)~</math> for various values of number <math>n</math> of iteration. Here,
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| <math>
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| \!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)
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| </math>
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| The iteration of <math>T</math> is implemented in the following way: :
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| <math>
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| \!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)
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| </math>
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| where
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| <math>\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ F(z)=\exp\Big( \exp\big( \ln(2)\, z\big)\Big)</math>
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| is [[superfunction]] for the transfer function
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| <math>T</math>
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| and
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| <math>
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| \!\!\!\!\!\!\!\!\!\!\ (4) ~ ~ ~G(z)=\ln\Big( \ln(z) \Big)/ \ln(2)
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| </math>
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| is the [[Abel Function]]. The thick lines represent
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| <math>
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| y=x^2
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| </math>and
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| <math>
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| y=\sqrt{x}</math>.
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| FIrst of them overlaps with <matm>y=T^1(x)</math> and corresponds to <math>n\!=\!1</math>, and the second overlaps with <math>y=T^{-1}(x)</math> and corresponds to <math>n\!=\!-1</math>.
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2012.11.11
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| |pub-country = Japan
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| |notes =
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| |versions = http://tori.ils.uec.ac.jp/TORI/index.php/File:PowIteT.jpg
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| }}
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| == Licensing ==
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| {{CC|by|3.0}}
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Latest revision as of 18:52, 11 March 2022
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