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imported>Dmitrii Kouznetsov
({{Image_Details|user |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, : $\!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)$ The iteration of $T$ is implemented in the following way: :$\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)$ wh...)
 
imported>Dmitrii Kouznetsov
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== Summary ==
== Summary ==
{{Image_Details|user
{{Image_Details|user
|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,  : $\!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)The iteration of $T$ is implemented in the following way:  :$\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)where  : $\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ F(z)=\exp\Big( \exp\big( \ln(2)\, z\big)\Big)$ is [[superfunction]] for the transfer function $T$ and :$\!\!\!\!\!\!\!\!\!\!\ (4) ~ ~ ~G(z)=\ln\Big( \ln(z) \Big)/ \ln(2)$ is the [[Abel Function]].  The thick lines represent $y=x^2$ and $y=\sqrt{x}$. FIrst of them overlaps with <matm>y=T^1(x)$ and corresponds to $c\!=\!1$, and the second overlaps with $y=T^{-1}(x)$ and corresponds to  $c\!=\!-1$.
|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,   
 
<math>
\!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)
</math>
 
The iteration of <math>T</math> is implemented in the following way:  :
 
<math>
\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)
</math>
 
where   
 
<math>\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ F(z)=\exp\Big( \exp\big( \ln(2)\, z\big)\Big)</math>
 
is [[superfunction]] for the transfer function  
<math>T</math>
and  
 
<math>
\!\!\!\!\!\!\!\!\!\!\ (4) ~ ~ ~G(z)=\ln\Big( \ln(z) \Big)/ \ln(2)
</math>
 
is the [[Abel Function]].  The thick lines represent  
<math>
y=x^2
</math>and  
<math>
y=\sqrt{x}</math>.  
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>.
|author      = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
|author      = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
|date-created = 2012.11.11
|date-created = 2012.11.11
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|versions    = http://tori.ils.uec.ac.jp/TORI/index.php/File:PowIteT.jpg
|versions    = http://tori.ils.uec.ac.jp/TORI/index.php/File:PowIteT.jpg
}}
}}
== Licensing ==
== Licensing ==
{{CC|by|3.0}}
{{CC|by|3.0}}

Revision as of 23:58, 24 November 2012

Summary

Title / Description


Explicit plot of th Iteration of the quadratic function, id est, power function for power 2; for various values of number of iteration. Here,

The iteration of is implemented in the following way:  :

where

is superfunction for the transfer function and

is the Abel Function. The thick lines represent and . FIrst of them overlaps with <matm>y=T^1(x)</math> and corresponds to , and the second overlaps with and corresponds to .

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Copyright © Dmitrii Kouznetsov.
See below for licence/re-use information.
Date created


2012.11.11
Country of first publication


Japan
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