Difference between revisions of "File:SimudoyaTb.png"

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{{oq|SimudoyaTb.png|Original file ‎(3,388 × 1,744 pixels, file size: 537 KB, MIME type: image/png)}}
⚫
Recovery of [[superfunction]] of the [[transfer function]] $T=\,$[[Doya function|Doya]]$_1$ by the [[Regular iteration]] method.
 
   
  +
Recovery of [[superfunction]] of the [[transfer function]]
⚫
$T$ can be expressed also through the [[LambertW]] function:
  
⚫
:$T(z)=\mathrm{Doya}_1(z)=\mathrm{LambertW}\Big( z~ \mathrm{e}^{z+1} \Big)$
  
   
  ⚫
\(T=\ \)[[Doya function|Doya]]\(_1\) by the [[Regular iteration]] method.
⚫
$T$ can be interpreted as a transfer function of the idealized laser amplifier; its superfunction $F$ describes the evolution of the normalized intensity along the direction of amplification.
  
   
  ⚫
\(T\) can be expressed also through the [[LambertW]] function:
⚫
Brown curve and the red curve represent the primary approximation of $F$ with one term and with two terms of the asymptotic expansion, respectively.
  
   
  ⚫
\(T(z)=\mathrm{Doya}_1(z)=\mathrm{LambertW}\Big( z~ \mathrm{e}^{z+1} \Big)\)
⚫
Green, cyan, blue and magenta represent the First, Second, Third and Fourth iteration of the approximation with 2 terms; the curves of corresponding dark colors represent the regular iterations of the primary approximation with single term.
  
   
−
For this example of the transfer function, the analytic representation of the [[superfunction]] $F~$ through the [[Tania function]] exist:
 +
\(T\) can be interpreted as a transfer function of the idealized laser amplifier.<br>
  ⚫
Its superfunction \(F\) describes the evolution of the normalized intensity along the direction of amplification.
⚫
: $F(x)=\mathrm{Tania}(x\!-\!1)=\mathrm{LambertW}(\exp(x))$
 
  +
  ⚫
Brown curve and the red curve represent the primary approximation of \(F\) with one term and with two terms of the asymptotic expansion, respectively.
  +
  ⚫
Green, cyan, blue and magenta curves represent the First, Second, Third and Fourth iteration of the approximation with 2 terms; the curves of corresponding dark colors represent the regular iterations of the primary approximation with single term.
  +
  +
For this example of the transfer function, the analytic representation of the [[superfunction]] \(F\) through the [[Tania function]] exist:
  +
  +
\[
  ⚫
F(x)=\mathrm{Tania}(x\!-\!1)=\mathrm{LambertW}(\exp(x))
  +
\]
  +
 
This superfunction is plotted with black curve.
  
This superfunction is plotted with black curve.
   
−
The primary approximations show good agreement while argument $x\!<\!-2$;
 +
The primary approximations show good agreement while argument \(x\!<\!-2\);
−
for larger values of x, transfer function $T$ should be iterated, roughtly, $n=x\!+\!2$ times, misplacing the argument for $n$; id est,
 +
for larger values of \(x\), transfer function \(T\) should be iterated, roughtly, \(n=x\!+\!2\) times, misplacing the argument for $\(n\); id est,
  +
\[
−
: $F(x)\approx T^n( \tilde F (x\!-\!n))$
 +
F(x)\approx T^n( \tilde F (x\!-\!n))\]
⚫
where $\tilde F$ is one of the primary approximations.
  
  +
  ⚫
Here \(\tilde F\) is one of the primary approximations.
  +
  +
==Superfunctions==
  +
The explicit plot above appears as Fig.6.2 at page 66 of book
  +
«[[Superfunctions]]»
  +
<ref>https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics - 2020/7/28
  +
</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
  +
</ref><br>
  +
in order to show how the [[regular iteration]] works, using the simple example with
  +
the easy transfer function that allows to trace each step of the evaluation of the [[superfunction]].
   
 
==C++ generator of curves==
  
==C++ generator of curves==
Line 69: Line 90:
 
getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
  
getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
 
}
  
}
  +
</pre>
−  
 
==Latex generator of labels==
  
==Latex generator of labels==
−
FIle [[simudoya.pdf]] should be generated with the code above in order to compile the [[Latex]] document below.
 +
File [[simudoya.pdf]] should be generated with the code above in order to compile the [[Latex]] document below.
  +
%<pre>
−  
−
% <nowiki> <br>
  
 
\documentclass[12pt]{article} %<br>
  
\documentclass[12pt]{article} %<br>
 
\usepackage{geometry} %<br>
  
\usepackage{geometry} %<br>
Line 125: Line 145:
 
\end{picture}} %<br>
  
\end{picture}} %<br>
 
\end{document} %<br>
  
\end{document} %<br>
−
% </nowiki> %<br>
 +
%</pre>
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
  +
==Keywords==
  +
«[[ArcTania]]»,
  +
«[[Doya function]]»,
  +
«[[Tania function]]»,
  +
«[[Superfunction]]»,
   
  +
[[Category:ArcTania]]
 
[[Category:Book]]
  
[[Category:Book]]
 
[[Category:BookPlot]]
  
[[Category:BookPlot]]

Revision as of 14:34, 19 August 2025


Recovery of superfunction of the transfer function

\(T=\ \)Doya\(_1\) by the Regular iteration method.

\(T\) can be expressed also through the LambertW function:

\(T(z)=\mathrm{Doya}_1(z)=\mathrm{LambertW}\Big( z~ \mathrm{e}^{z+1} \Big)\)

\(T\) can be interpreted as a transfer function of the idealized laser amplifier.
Its superfunction \(F\) describes the evolution of the normalized intensity along the direction of amplification.

Brown curve and the red curve represent the primary approximation of \(F\) with one term and with two terms of the asymptotic expansion, respectively.

Green, cyan, blue and magenta curves represent the First, Second, Third and Fourth iteration of the approximation with 2 terms; the curves of corresponding dark colors represent the regular iterations of the primary approximation with single term.

For this example of the transfer function, the analytic representation of the superfunction \(F\) through the Tania function exist:

\[ F(x)=\mathrm{Tania}(x\!-\!1)=\mathrm{LambertW}(\exp(x)) \]

This superfunction is plotted with black curve.

The primary approximations show good agreement while argument \(x\!<\!-2\); for larger values of \(x\), transfer function \(T\) should be iterated, roughtly, \(n=x\!+\!2\) times, misplacing the argument for $\(n\); id est, \[ F(x)\approx T^n( \tilde F (x\!-\!n))\]

Here \(\tilde F\) is one of the primary approximations.

Superfunctions

The explicit plot above appears as Fig.6.2 at page 66 of book «Superfunctions» [1][2]
in order to show how the regular iteration works, using the simple example with the easy transfer function that allows to trace each step of the evaluation of the superfunction.

C++ generator of curves

// FIles ado.cin and doya.cin should be loaded to the current directory in order to compile the C++ code below: 

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
DB ArcTania(DB x){ return x+log(x)-1.;}
#include "ado.cin"
#include "doya.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
main(){ int j,k,m,n; DB x,y,t,e, a; z_type z;
FILE *o;o=fopen("simudoya.eps","w");ado(o,808,408);
fprintf(o,"404 4 translate\n 100 100 scale\n");
for(m=-4;m<5;m++){ M(m,0)L(m,4)}
for(n=0;n<5;n++){ M(-4,n)L(4,n)}
fprintf(o,"2 setlinecap\n");
fprintf(o,".006 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 1 setlinecap\n");
// for(n=0;n<380;n++){y=.02+.01*n; x=ArcTania(y); if(n==0)M(x,y) else L(x,y)} fprintf(o,".03 W 0 1 0 RGB S\n");
for(n=0;n<82;n++){x=-4.05+.1*n;z=x;y=Re(Tania(z)); if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 0 0 RGB S\n");
//
for(n=0;n<45;n++){x=-4+.1*n;e=exp(x+1.); y=e;                       ; if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W .4 0 0 RGB S\n");
for(n=0;n<70;n++){x=-4+.04*n;e=exp(x+1.); y=e*(1.-e);                ; if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 1 0 0 RGB S\n");
//
for(n=0;n<53;n++){x=-4+.1*n;e=exp(x+0.); y=e       ; y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 .4 0 RGB S\n");
for(n=0;n<190;n++){x=-4+.02*n;e=exp(x+0.); y=e*(1.-e); y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 1 0 RGB S\n");
//
for(n=0;n<61;n++){x=-4+.1*n;e=exp(x-1.); y=e       ;DO(m,2)y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 .4 0.4 RGB S\n");
for(n=0;n<240;n++){x=-4+.02*n;e=exp(x-1.); y=e*(1.-e);DO(m,2)y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 1 1 RGB S\n");
//
for(n=0;n<67;n++){x=-4+.1*n;e=exp(x-2.); y=e       ;DO(m,3)y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 0 0.4 RGB S\n");
for(n=0;n<288;n++){x=-4+.02*n;e=exp(x-2.); y=e*(1.-e);DO(m,3)y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 0 0 1 RGB S\n");
//
for(n=0;n<73;n++){x=-4+.1*n;e=exp(x-3.); y=e       ;DO(m,4)y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W .4 0 0.4 RGB S\n");
for(n=0;n<340;n++){x=-4+.02*n;e=exp(x-3.); y=e*(1.-e);DO(m,4)y=Re(Doya(1.,y));if(n==0)M(x,y) else L(x,y)} fprintf(o,".01 W 1 0 1 RGB S\n");
//
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
       system("epstopdf simudoya.eps");
       system(    "open simudoya.pdf"); //these 2 commands may be specific for macintosh
getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
}

Latex generator of labels

File simudoya.pdf should be generated with the code above in order to compile the Latex document below.

%
\documentclass[12pt]{article} %<br>
\usepackage{geometry}  %<br>
\usepackage{graphicx} %<br>
\usepackage{hyperref} %<br>
\usepackage{rotating} %<br>
\paperwidth  1632pt %<br>
\paperheight 840pt %<br>
\topmargin -102pt %<br>
\oddsidemargin -58pt %<br>
\textwidth 1825pt %<br>
\textheight 1470pt %<br>
\newcommand \sx {\scalebox} %<br>
\newcommand \ing {\includegraphics} %<br>
\newcommand \tet {\mathrm{tet}} %<br>
\newcommand \pen {\mathrm{pen}} %<br>
\newcommand \bC {\mathbb C} %<br>
\newcommand \fac {\mathrm {Factorial}} %<br>
\newcommand \rme {\mathrm e} %<br>
\newcommand \rmi {\mathrm i} %<br>
\newcommand \ds {\displaystyle} %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\pagestyle{empty} %<br>
\begin{document} %<br>
\parindent 0pt %<br>
\sx{2}{\begin{picture}(600,404) %<br>
\put(0,0){\ing{simudoya}} %<br>
\put(-7,300){\sx{1.7}{$3$}} %<br>
\put(-7,200){\sx{1.7}{$2$}} %<br>
\put(-7,100){\sx{1.7}{$1$}} %<br>
\put(90,-11){\sx{1.7}{$-2$}} %<br>
\put(190,-11){\sx{1.7}{$-1$}} %<br>
\put(301,-11){\sx{1.7}{$0$}} %<br>
\put(401,-11){\sx{1.7}{$1$}} %<br>
\put(501,-11){\sx{1.7}{$2$}} %<br>
\put(601,-11){\sx{1.7}{$3$}} %<br>
\put(701,-11){\sx{1.7}{$4$}} %<br>
\put(798,-11){\sx{1.7}{$x$}} %<br>
\put(366,206){\rot{65}\sx{2}{$\exp(x)$}\ero} %<br>
\put(436,206){\rot{66}\sx{2}{$T(\exp(x\!-\!1))$}\ero} %<br>
\put(530,266){\rot{66}\sx{2}{$T^2(\exp(x\!-\!2))$}\ero} %<br>
\put(580,266){\rot{60}\sx{2}{$T^3(\exp(x\!-\!3))$}\ero} %<br>
\put(620,266){\rot{55}\sx{2}{$T^4(\exp(x\!-\!4))$}\ero} %<br>
\put(733,322){\rot{41}\sx{2.1}{$F(x)$}\ero} %<br>
\put(548,170){\sx{1.7}{$T^4(\exp(x\!-\!4)\!-\!\exp(2(x\!-\!4)))$}} %<br>
\put(508,116){\sx{1.7}{$T^3(\exp(x\!-\!3)-\exp(2(x\!-\!3)))$}} %<br>
\put(418,64){\sx{1.7}{$T^2(\exp(x\!-\!2)-\exp(2(x\!-\!2)))$}} %<br>
\put(280,35){\sx{1.7}{$T(\exp(x-1)-\exp(2(x\!-\!1)))$}} %<br>
\put(160,11){\sx{1.7}{$\exp(x)-\exp(2x))$}} %<br>
\end{picture}} %<br>
\end{document} %<br>
%

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

Keywords

«ArcTania», «Doya function», «Tania function», «Superfunction»,

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