Difference between revisions of "File:ZexIteT.jpg"

Explicit plot of iterations of function Zex

\( y=\mathrm{zex}^n(x)\)

for various number $n$ of iterates. Here, \(\ \mathrm{zex}(z)\!=\!z\exp(z)\ \). The non-integer iterates are implemented through the superfunction of zex and its abelfunction, named SuZex and AuZex, as follows:

\( \mathrm{zex}^n(x)=\mathrm{SuZex}\Big(n+\mathrm{AuZex}(x)\Big) \)

This plot appears as Fig.11.8 at page 147 of book «Superfunctions», 2020 ^[1][2]
as a verification, a support of the pretentious claim that with formalism of Superfunctions, the Author can iterate any growing real-holomorphic function, and the number \(n\) of the iterate has no need to be integer.

C++ generator of curves

// Files ado.cin, Tania.cin, LambertW.cin, SuZex.cin, AuZex.cin should be loaded to the working directory in order to compile the 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.)

 #include "Tania.cin" // need for LambertW
 #include "LambertW.cin" // need for AuZex
 #include "SuZex.cin"
 #include "AuZex.cin"
 #include "ado.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, p,q, t; z_type z,c,d;  FILE *o;o=fopen("ZexIte.eps","w");  ado(o,1204,1204);
 fprintf(o,"2 2 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 for(n=0;n<13;n++) {M(0,n)L(12,n)}
 for(m=0;m<13;m++) {M(m,0)L(m,12)}
 M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)  fprintf(o,".01 W S\n");
 
 DO(m,700){x=.01 +.02*m; y=Re(LambertW(LambertW(x)));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".033 W 1 0 1 RGB S\n");
 DO(m,700){x=.01 +.02*m; y=Re(LambertW(x));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".04 W 1 0 1 RGB S\n");
 M(0,0) L(12.03,12.03) fprintf(o,".03 W 0 1 0 RGB S\n");
 DO(m,700){x=.01 +.02*m; y=Re(zex(x));     if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".04 W 0 1 0 RGB S\n");
 DO(m,700){x=.01 +.02*m; y=Re(zex(zex(x)));     if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".033 W 0 1 0 RGB S\n");

 for(n=-10;n<11;n++){
   DO(m,700){x=.01 +.02*m; y=Re(auzex(x)); y=Re(suzex(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;}  
   fprintf(o,".023 W 0 0 0 RGB S\n");
                    }
 fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
       system("epstopdf ZexIte.eps"); 
       system(    "open ZexIte.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex generator of labels

% file ZexIte.pdf should be generated with the code above in order to compile the Latex document below. 
% Copyleft 2012 by Dmitrii Kouznetsov <br> %
\documentclass[12pt]{article}  % <br> 
\usepackage{geometry}  % <br> 
\usepackage{graphicx} % <br> 
\usepackage{rotating} % <br> 
\paperwidth 1208pt % <br> 
\paperheight 1208pt % <br> 
\topmargin -103pt % <br> 
\oddsidemargin -73pt % <br> 
\textwidth 1404pt % <br> 
\textheight 1404pt % <br> 
\pagestyle {empty} % <br> 
\newcommand \sx {\scalebox} % <br> 
\newcommand \rot {\begin{rotate}} % <br> 
\newcommand \ero {\end{rotate}} % <br> 
\newcommand \ing {\includegraphics} % <br> 
\parindent 0pt%  <br> 
\pagestyle{empty} % <br> 
\begin{document}  % <br> 
\begin{picture}(1202,1202) % <br> 
%\put(10,10){\ing{IterPowPlot}} % <br> 
%\put(10,10){\ing{IterEq2plot}} % <br> 
\put(0,0){\ing{ZexIte}} % <br> 
\put(11,1184){\sx{4.4}{$y$}} % <br> 
\put(04,1090){\sx{4}{$11$}} % <br> 
\put(04,990){\sx{4}{$10$}} % <br> 
\put(11,890){\sx{4}{$9$}} % <br> 
\put(11,790){\sx{4}{$8$}} % <br> 
\put(11,690){\sx{4}{$7$}} % <br> 
\put(11,590){\sx{4}{$6$}} % <br> 
\put(11,490){\sx{4}{$5$}} % <br> 
\put(11,390){\sx{4}{$4$}} % <br> 
\put(11,290){\sx{4}{$3$}} % <br> 
\put(11,190){\sx{4}{$2$}} % <br> 
\put(11,090){\sx{4}{$1$}} % <br> 
 % <br> 
\put(91,6){\sx{4}{$1$}} % <br> 
\put(191,6){\sx{4}{$2$}} % <br> 
\put(291,6){\sx{4}{$3$}} % <br> 
\put(391,6){\sx{4}{$4$}} % <br> 
\put(492,6){\sx{4}{$5$}} % <br> 
\put(592,6){\sx{4}{$6$}} % <br> 
\put(693,6){\sx{4}{$7$}} % <br> 
\put(794,6){\sx{4}{$8$}} % <br> 
\put(894,6){\sx{4}{$9$}} % <br> 
\put(982,6){\sx{4}{$10$}} % <br> 
\put(1082,6){\sx{4}{$11$}} % <br> 
\put(1180,6){\sx{4.4}{$x$}} % <br> 
 % <br> 
 
\put(116,1058){\sx{5}{\rot{88}$n\!=\!2$\ero}} % <br> 
\put(182,1058){\sx{5}{\rot{88}$n\!=\!1$\ero}} % <br> 
\put(141,706){\sx{4.5}{\rot{86}$y\!=\!\mathrm{zex}(x)$\ero}} % <br> 
%
\put(512,1030){\sx{5}{\rot{72}$n\!=\!0.3$\ero}} % <br> 
\put(629,1032){\sx{5}{\rot{64}$n\!=\!0.2$\ero}} % <br> 
\put(804,1037){\sx{5}{\rot{56}$n\!=\!0.1$\ero}} % <br> 
 % <br> 
\put(1072,1052){\sx{5}{\rot{44}$n\!=\!0$\ero}} % <br> 
\put(772,752){\sx{5}{\rot{44}$y\!=\!x$\ero}} % <br> 
% <br> 
\put(1028,762){\sx{5}{\rot{32}$n\!=\!-0.1$\ero}} % <br> 
\put(1014,590){\sx{5}{\rot{23}$n\!=\!-0.2$\ero}} % <br> 
\put(1006,470){\sx{5}{\rot{17}$n\!=\!-0.3$\ero}} % <br> 
%
\put(510,108){\sx{4.3}{\rot{3}$y\!=\!\mathrm{LambertW}(x)$\ero}} % <br> 
\put(1010,142){\sx{5}{\rot{2}$n\!=\!-1$\ero}} % <br> 
\put(1010,83){\sx{5}{\rot{1}$n\!=\!-2$\ero}} % <br> 
%<br>
\end{picture} % <br> 
\end{document} % <br> 

References

Keywords

«Abelfunction», «ArcLambertW», «ArcZex», «AuZex», «Exotic iteration», «Fixed point», «Iterate», «Iteration», «LambertW», «ProductLog», «Superfunction», «Superfunctions», «SuZex», «Transfer function», «Transferfunction», «Zex»,

«Суперфункции»,