Difference between revisions of "File:SuZexPlot511T.jpg"

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{{oq|SuZexPlot511T.jpg|Original file ‎(1,055 × 2,292 pixels, file size: 188 KB, MIME type: image/jpeg) }}
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Importing image file
  
  +
  +
[[Explicit plot]] of function [[SuZex]] in comparison to its transfer function [[zex]]\((z)\!=\!z\,\exp(z)\):
  +
  +
\(y\!=\!\mathrm{SuZex}(x)\) and
  +
  +
\(y\!=\!\mathrm{zex}(x)\)
  +
  +
versus \(x\)
  +
  +
This plot is used as Fig.11.5 at page 142 of book «[[Superfunctions]]», 2020
  +
<ref name="a">
  +
https://www.amazon.com/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. For a given transferfunction T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfunction is inverse of F. In particular, superfunctions of factorial, exp, sin are suggested. The Holomorphic extensions of the logistic sequence and those of the Ackermann functions are considered. Among ackermanns, the tetration (mainly to the base b>1) and natural pentation (to base b=e) are presented. The efficient algorithm for the evaluation of superfunctions and abelfunctions are described. The graphics and complex maps are plotted. The possible applications are discussed. Superfunctions significantly extend the set of functions, that can be used in scientific research and technical design.
  +
</ref><ref name="t">
  +
https://mizugadro.mydns.jp/BOOK/468.pdf
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref><br>
  +
in order to compare functions [[zex]] and [[SuZex]].
  +
  +
==[[C++]] generator of curves==
  +
// Files [[ado.cin]]. [[SuZexCoef20.cin]], [[SuZexTay2008co.cin]], [[SuZexTay0co.cin]], [[SuZex.cin]]
  +
//should be loaded to the working directory in order to compile the code below.
  +
<pre>
  +
#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.)
  +
  +
/*
  +
z_type zex(z_type z) { return z*exp(z);}
  +
  +
z_type suzexo(z_type z){ int n,m; z_type L=log(-z); z_type c[21]; z_type s;
  +
#include "SuZexCoef20.cin"
  +
c[0]=-1.; c[1]=.5*L;
  +
for(n=2;n<21;n++){ s=a[n][n];
  +
for(m=n-1; m>=0; m--){ s*=L; s+=a[n][m]; }
  +
c[n]=s; }
  +
s=c[20]/z; for(n=19;n>=0;n--){ s+=c[n]; s/=z;}
  +
return s; }
  +
  +
z_type suzex2008t12(z_type z){ int n,m=80; z_type s;
  +
#include "SuZexTay2008co.cin"
  +
s=c[m]; for(n=m-1;n>=0;n--){ s*=z; s+=c[n];}
  +
return s;}
  +
  +
z_type SuZexTay0(z_type z){ int n,m=96; z_type s;
  +
#include "SuZexTay0co.cin"
  +
s=SuZexTay0co[m]; for(n=m-1;n>=0;n--){ s*=z; s+=SuZexTay0co[n];}
  +
return s;}
  +
  +
z_type suzex(z_type z){int m,n; z_type s;
  +
if( abs(z) < 1.6 ) return SuZexTay0(z) ; // I made the Taylor expansion for this case
  +
if( Re(z)>0 && fabs(Im(z))<1.5){n=int(Re(z)+.5); s=SuZexTay0(z-(0.+n)); DO(m,n) s=zex(s); return s;}
  +
z+=-1.1259817765745026; // WARNING! ARGUEMENT CHANGES ITS VALUE!
  +
if( abs(z+12.) < 8.1 ) return suzex2008t12(z+12.) ; // I made the Taylor expansion for this case
  +
if( Re(z)<-12. || fabs(Im(z)) > 8. ) return suzexo(z); // definitely, |z|>8
  +
n= int(Re(z)+12.); s=suzex2008t12(z+12.-(0.+n)); DO(m,n) s=zex(s); return s;
  +
}
  +
// Copyleft 2012 by Dmitrii Kouznetsov
  +
*/
  +
  +
#include "SuZex.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, cu,cd;
  +
DB x1=-1.1259817765745026; //DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
  +
FILE *o;o=fopen("SuZexPlot511.eps","w"); ado(o,504,1104);
  +
fprintf(o,"303 101 translate\n 100 100 scale\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
for(m=-3;m<3;m++) {M(m,-1)L(m,16)}
  +
for(n=-1;n<17;n++) {M(-3,n)L(2,n)}
  +
M(0,M_E)L(1,M_E)
  +
M(0,-1./M_E)L(-1,-1./M_E)
  +
fprintf(o,".006 W S\n");
  +
//
  +
DO(m,130){x=-3.02 +.1*m; y=Re(suzex(x)); if(m==0) M(x,y) else L(x,y) if(y>10) break;} fprintf(o,".03 W 0 0 1 RGB S\n");
  +
// DO(m,130){x=-10.+.1*m; y=-1./(x+x1); if(m==0) M(x,y) else L(x,y) if(y>10) break;} fprintf(o,".01 W 0 0 0 RGB S\n");
  +
// DO(m,130){x=-2.+.1*m; y=exp(x); if(m==0) M(x,y) else L(x,y) if(y>10) break;} fprintf(o,".01 W .5 0 0 RGB S\n");
  +
DO(m,130){x=-3.02+.1*m; y=Re(zex(x)); if(m==0) M(x,y) else L(x,y) if(y>10) break;} fprintf(o,".01 W 0 0 0 RGB S\n");
  +
//
  +
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o);
  +
system("epstopdf SuZexPlot511.eps");
  +
system( "open SuZexPlot511.pdf"); //for macintosh
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
  +
</pre>
  +
==[[Latex]] generator of labels==
  +
%<pre>
  +
% file SuZexPlot.pdf should be generated with the code above in order to compile the Latex document below. %<br>
  +
% Copyleft 2012 by Dmitrii Kouznetsov <br> %
  +
\documentclass[12pt]{article} % <br>
  +
\usepackage{geometry} % <br>
  +
\usepackage{graphicx} % <br>
  +
\usepackage{rotating} % <br>
  +
\paperwidth 508pt % <br>
  +
\paperheight 1104pt % <br>
  +
\topmargin -96pt % <br>
  +
\oddsidemargin -81pt % <br>
  +
\textwidth 1200pt % <br>
  +
\textheight 1200pt % <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}(1202,1102) % <br>
  +
%\put(10,10){\ing{ExpQ2plot}} % <br>
  +
\put(322,1090){\sx{4}{$y$}} % <br>
  +
\put(292,372){\sx{4}{$\mathrm e$}} % <br>
  +
\put(315, 62){\sx{4}{$-\!1/\mathrm e$}} % <br>
  +
\put(325,998){\sx{4}{$9$}} % <br>
  +
\put(325,898){\sx{4}{$8$}} % <br>
  +
\put(325,798){\sx{4}{$7$}} % <br>
  +
\put(325,698){\sx{4}{$6$}} % <br>
  +
\put(325,598){\sx{4}{$5$}} % <br>
  +
\put(325,498){\sx{4}{$4$}} % <br>
  +
\put(325,398){\sx{4}{$3$}} % <br>
  +
\put(325,298){\sx{4}{$2$}} % <br>
  +
\put(325,198){\sx{4}{$1$}} % <br>
  +
%\put(325,098){\sx{4}{$0$}} % <br>
  +
% <br>
  +
\put(078,116){\sx{4}{$-\!2$}} % <br>
  +
\put(178,116){\sx{4}{$-\!1$}} % <br>
  +
\put(304,116){\sx{4}{$0$}} % <br>
  +
\put(404,116){\sx{4}{$1$}} % <br>
  +
\put(490,116){\sx{4}{$x$}} % <br>
  +
% <br>
  +
\put(12,188){\sx{4}{\rot{0}$y\!=\!\mathrm{SuZex}(x)$\ero}} % <br>
  +
\put(358,162){\sx{4}{\rot{0}$y\!=\!x\mathrm{e}^x$\ero}} % <br>
  +
%\put(10,10){\ing{ExpQ2plot}} % <br>
  +
\put(10,10){\ing{SuZexPlot511}} % <br>
  +
\end{picture} % <br>
  +
\end{document} % <br>
  +
%</pre>
  +
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
==Keywords==
  +
  +
«[[ArcLambertW]]»,
  +
«[[Exotic iteration]]»,
  +
«[[Fixed point]]»,
  +
«[[LambertW]]»,
  +
«[[ProductLog]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[SuZex]]»,
  +
«[[Zex]]»,
  +
  +
«[[Суперфункции]]»,
  +
  +
[[Category:ArcZex]]
  +
[[Category:Book]]
  +
[[Category:BookE]]
  +
[[Category:BookPlot]]
  +
[[Category:BookPlotE]]
  +
[[Category:C++]]
  +
[[Category:Exotic iteration]]
  +
[[Category:Explicit plot]]
  +
[[Category:Inverse functions]]
  +
[[Category:Iterate]]
  +
[[Category:LambertW]]
  +
[[Category:LambertW function]]
  +
[[Category:Laser science]]
  +
[[Category:Latex]]
  +
[[Category:Mahtematics of Computation]]
  +
[[Category:Special function]]
  +
[[Category:Superfunction]]
  +
[[Category:SuZex]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]
  +
[[Category:Tania function]]
  +
[[Category:Transfer function]]
  +
[[Category:zex]]

Latest revision as of 16:25, 26 August 2025


Explicit plot of function SuZex in comparison to its transfer function zex\((z)\!=\!z\,\exp(z)\):

\(y\!=\!\mathrm{SuZex}(x)\) and

\(y\!=\!\mathrm{zex}(x)\)

versus \(x\)

This plot is used as Fig.11.5 at page 142 of book «Superfunctions», 2020 [1][2]
in order to compare functions zex and SuZex.

C++ generator of curves

// Files ado.cin. SuZexCoef20.cin, SuZexTay2008co.cin, SuZexTay0co.cin, SuZex.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.)

/*
 z_type zex(z_type z) { return z*exp(z);}

 z_type suzexo(z_type z){ int n,m; z_type L=log(-z); z_type c[21]; z_type s;
 #include "SuZexCoef20.cin"
 c[0]=-1.; c[1]=.5*L;
 for(n=2;n<21;n++){ s=a[n][n]; 
                    for(m=n-1; m>=0; m--){ s*=L; s+=a[n][m]; }
                    c[n]=s; }
 s=c[20]/z; for(n=19;n>=0;n--){ s+=c[n]; s/=z;}
 return s; }

  z_type suzex2008t12(z_type z){ int n,m=80; z_type s;
  #include "SuZexTay2008co.cin"
  s=c[m]; for(n=m-1;n>=0;n--){ s*=z; s+=c[n];}
  return s;}

  z_type SuZexTay0(z_type z){ int n,m=96; z_type s;
  #include "SuZexTay0co.cin"
  s=SuZexTay0co[m]; for(n=m-1;n>=0;n--){ s*=z; s+=SuZexTay0co[n];}
  return s;}

 z_type suzex(z_type z){int m,n; z_type s;
 if( abs(z) < 1.6 )                   return SuZexTay0(z) ; // I made the Taylor expansion for this case
 if( Re(z)>0 && fabs(Im(z))<1.5){n=int(Re(z)+.5); s=SuZexTay0(z-(0.+n)); DO(m,n) s=zex(s); return s;}
  z+=-1.1259817765745026;     // WARNING! ARGUEMENT CHANGES ITS VALUE!
 if( abs(z+12.) < 8.1 )               return suzex2008t12(z+12.) ; // I made the Taylor expansion for this case
 if( Re(z)<-12. || fabs(Im(z)) > 8. ) return suzexo(z); // definitely, |z|>8
 n= int(Re(z)+12.); s=suzex2008t12(z+12.-(0.+n)); DO(m,n) s=zex(s); return s;
 } 
 // Copyleft 2012 by Dmitrii Kouznetsov
 */

 #include "SuZex.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, cu,cd;
 DB x1=-1.1259817765745026; //DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
 FILE *o;o=fopen("SuZexPlot511.eps","w");  ado(o,504,1104);
 fprintf(o,"303 101 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 for(m=-3;m<3;m++) {M(m,-1)L(m,16)}
 for(n=-1;n<17;n++) {M(-3,n)L(2,n)}
 M(0,M_E)L(1,M_E)
 M(0,-1./M_E)L(-1,-1./M_E)
 fprintf(o,".006 W S\n");
 //
 DO(m,130){x=-3.02 +.1*m; y=Re(suzex(x)); if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".03 W 0 0 1 RGB S\n");
 // DO(m,130){x=-10.+.1*m; y=-1./(x+x1);   if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".01 W 0 0 0 RGB S\n");
 // DO(m,130){x=-2.+.1*m; y=exp(x);       if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".01 W .5 0 0 RGB S\n");
  DO(m,130){x=-3.02+.1*m; y=Re(zex(x));    if(m==0) M(x,y) else L(x,y) if(y>10) break;}  fprintf(o,".01 W 0 0 0 RGB S\n");
 //
 fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
       system("epstopdf SuZexPlot511.eps"); 
       system(    "open SuZexPlot511.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex generator of labels

%
% file SuZexPlot.pdf should be generated with the code above in order to compile the Latex document below. %<br>
% Copyleft 2012 by Dmitrii Kouznetsov <br> %
\documentclass[12pt]{article}  % <br> 
\usepackage{geometry}  % <br> 
\usepackage{graphicx} % <br> 
\usepackage{rotating} % <br> 
\paperwidth 508pt % <br> 
\paperheight 1104pt % <br> 
\topmargin -96pt % <br> 
\oddsidemargin -81pt % <br> 
\textwidth 1200pt % <br> 
\textheight 1200pt % <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}(1202,1102) % <br> 
%\put(10,10){\ing{ExpQ2plot}} % <br> 
\put(322,1090){\sx{4}{$y$}} % <br> 
\put(292,372){\sx{4}{$\mathrm e$}} % <br>
\put(315, 62){\sx{4}{$-\!1/\mathrm e$}} % <br>  
\put(325,998){\sx{4}{$9$}} % <br> 
\put(325,898){\sx{4}{$8$}} % <br> 
\put(325,798){\sx{4}{$7$}} % <br> 
\put(325,698){\sx{4}{$6$}} % <br> 
\put(325,598){\sx{4}{$5$}} % <br> 
\put(325,498){\sx{4}{$4$}} % <br> 
\put(325,398){\sx{4}{$3$}} % <br> 
\put(325,298){\sx{4}{$2$}} % <br> 
\put(325,198){\sx{4}{$1$}} % <br> 
%\put(325,098){\sx{4}{$0$}} % <br> 
 % <br> 
\put(078,116){\sx{4}{$-\!2$}} % <br> 
\put(178,116){\sx{4}{$-\!1$}} % <br> 
\put(304,116){\sx{4}{$0$}} % <br> 
\put(404,116){\sx{4}{$1$}} % <br> 
\put(490,116){\sx{4}{$x$}} % <br> 
 % <br> 
\put(12,188){\sx{4}{\rot{0}$y\!=\!\mathrm{SuZex}(x)$\ero}} % <br> 
\put(358,162){\sx{4}{\rot{0}$y\!=\!x\mathrm{e}^x$\ero}} % <br> 
%\put(10,10){\ing{ExpQ2plot}} % <br> 
\put(10,10){\ing{SuZexPlot511}} % <br> 
\end{picture} % <br> 
\end{document} % <br> 
%

References

  1. ↑ https://www.amazon.com/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020. Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. For a given transferfunction T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfunction is inverse of F. In particular, superfunctions of factorial, exp, sin are suggested. The Holomorphic extensions of the logistic sequence and those of the Ackermann functions are considered. Among ackermanns, the tetration (mainly to the base b>1) and natural pentation (to base b=e) are presented. The efficient algorithm for the evaluation of superfunctions and abelfunctions are described. The graphics and complex maps are plotted. The possible applications are discussed. Superfunctions significantly extend the set of functions, that can be used in scientific research and technical design.
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

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