Difference between revisions of "File:TraitT.jpg"

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{{oq|TraitT.jpg|Original file ‎(1,258 × 1,258 pixels, file size: 604 KB, MIME type: image/jpeg)|480}}
−
[[Iterate]]s of the [[Trappmann function]], $~\mathrm{tra}(z)=x+\mathrm e^z$.
  
   
  +
Fig.20.12 from page 288 of book «[[Superfunctions]]»<ref>
−
$y=\mathrm{tra}^n(x)=\mathrm{hen}\Big(n+\mathrm{ahe}(x)\Big)$
  
  +
https://mizugadro.mydns.jp/BOOK/468.pdf
  +
D.Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>, 2020.
   
  +
The same picture is used also as Рис.20.12 at page 298 of the Russian version «[[Суперфункции]]»
−
versus $x$ for various values of $n$.
  
  +
<ref>
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
  +
https://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014
  +
</ref>, 2014.
   
  +
And also as Figure 5 at page 6537 of article [[Entire Function with Logarithmic Asymptotic]]
−
The [[Henryk function]] hen and the [[ArcHenryk]] ahe are [[superfunction]] and [[Abel function]] of the [[Trappmann function]]. They can be expressed through [[superfunction]] and [[Abel function]] of function [[zex]].
  
  +
<ref>
  +
https://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf <br>
  +
https://mizugadro.mydns.jp/PAPERS/2013hikari.pdf D.Kouznetsov. [[Entire Function with Logarithmic Asymptotic]]. [[Applied Mathematical Sciences]], Vol. 7, 2013, no. 131, 6527 - 6541
  +
</ref>
  +
at [[Applied Mathematical Sciences]], 2013.
   
  +
The image shows the iterates of the [[Trappmann function]] \( \mathrm{tra} = z \mapsto z+\exp(z) \):
−
==[[C++]] generator of curves==
  
  +
\( y=\mathrm{tra}^n(y) \) versus \(x\)
  +
for various values of number \(n\) of iterate.
   
  +
The \(n\)th iterate of the [[Trappmann function]] tra is expressed through its [[superfunction]] [[SuTra]]
−
// Files [[ado.cin]], [[Tania.cin]], [[LambertW.cin]], [[SuZex.cin]], [[AuZex.cin]]
  
  +
and the [[Abelfunction]] [[AuTra]]:
−
//should be loaded to the working directory in order to compile the code below.
  
  +
\[\mathrm{tra}^n(z)= \mathrm{SuTra}(n+\mathrm{AuTra}(z))
  +
\]
  +
The number \(n\) of iterate in this expression has no need to be integer.
  +
  +
==Implementation==
  +
Namely this picture is generated using the representation of functions [[SuTra]] and [[AuTra]]
  +
through the
  +
[[Doya function]],
  +
[[LambertW]] function] ([[Tania function]]),
  +
[[SuZex]] function.<br>
  +
So, files
  +
[[doya.cin]]
  +
[[LambertW.cin]]
  +
[[SuZex.cin]] and
  +
[[AuZex.cin]]
  +
should be loaded in order to compile the code below.
   
  +
==C++==
−
#include <math.h>
  
  +
<pre>
−
#include <stdio.h>
  
−
#include <stdlib.h>
 +
#include <math.h>
  +
#include <stdio.h>
−
#define DB double
  
  +
#include <stdlib.h>
−
#define DO(x,y) for(x=0;x<y;x++)
  
  +
#define DB double
−
using namespace std;
  
  +
#define DO(x,y) for(x=0;x<y;x++)
−
#include<complex>
  
  +
//using namespace std;
−
typedef complex<double> z_type;
  
  +
#include<complex>
−
#define Re(x) x.real()
  
  +
typedef std::complex<double> z_type;
−
#define Im(x) x.imag()
  
−
#define I z_type(0.,1.)
 +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
//#include "Tania.cin" // need for LambertW
  +
#include "doya.cin" // need for LambertW
  +
#include "LambertW.cin" // need for AuZex
  +
#include "SuZex.cin"
  +
#include "AuZex.cin"
  +
z_type tra(z_type z){ return exp(z)+z;}
  +
z_type F(z_type z){ return log(suzex(z));}
  +
z_type G(z_type z){ return auzex(exp(z));}
  +
#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);
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("trait.eps","w"); ado(o,604,604);
  +
fprintf(o,"302 302 translate\n 100 100 scale\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
for(n=-3;n<4;n++) {M(-3,n)L(3,n)}
  +
for(m=-3;m<4;m++) {M(m,-3)L(m,3)}
  +
// M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)
  +
fprintf(o,".004 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 0 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 .5 0 RGB S\n");
  +
// M(0,0) L(12.03,12.03) fprintf(o,".04 W 0 1 0 RGB S\n");
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(x)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(x))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(x)))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(x))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(x)))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(x))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(tra(x)))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(tra(tra(x)))))))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  +
fprintf(o,".022 W 0 1 0 RGB S\n");
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(y)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(y))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(y)))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(y))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(y)))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(y))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(tra(y)))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(tra(tra(y)))))))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
fprintf(o,".022 W 1 0 1 RGB S\n");
  +
M(-3,-3)L(3,3) fprintf(o,".022 W 0 .6 1 RGB S\n");
  +
for(n=-20;n<21;n+=1){
  +
DO(m,700){x=-3.01 +.02*m; y=Re(G(x)); y=Re(F(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  +
fprintf(o,".01 W 0 0 0 RGB S\n");
  +
}
  +
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o);
  +
system("epstopdf trait.eps");
  +
system( "open trait.pdf"); //for macintosh
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
   
  +
</pre>
−
#include "Tania.cin" // need for LambertW
  
  +
==Latex==
−
#include "LambertW.cin" // need for AuZex
  
  +
<pre>
−
#include "SuZex.cin"
  
  +
\documentclass[12pt]{article}
−
#include "AuZex.cin"
  
  +
\usepackage{geometry}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\paperwidth 606pt
  +
\paperheight 606pt
  +
\topmargin -105pt
  +
\oddsidemargin -73pt
  +
\textwidth 1100pt
  +
\textheight 1100pt
  +
\pagestyle {empty}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\parindent 0pt
  +
\pagestyle{empty}
  +
\begin{document}
  +
\begin{picture}(602,602)
  +
%\put(10,10){\ing{PowPlo}}
  +
\put(0,0){\ing{TraIt}}
  +
\put(311,590){\sx{2.5}{$y$}}
  +
\put(311,495){\sx{2.4}{$2$}}
  +
\put(311,395){\sx{2.4}{$1$}}
  +
\put(311,295){\sx{2.4}{$0$}}
  +
\put(307,194){\sx{2.4}{$-1$}}
  +
\put(307,093.4){\sx{2.4}{$-2$}}
   
  +
\put(083,308){\sx{2.4}{$-2$}}
−
z_type tra(z_type z){ return exp(z)+z;}
  
  +
\put(183,308){\sx{2.4}{$-1$}}
−
z_type F(z_type z){ return log(suzex(z));}
  
  +
\put(297,308){\sx{2.4}{$0$}}
−
z_type G(z_type z){ return auzex(exp(z));}
  
  +
\put(397,308){\sx{2.4}{$1$}}
  +
\put(497,308){\sx{2.4}{$2$}}
  +
\put(590,308){\sx{2.5}{$x$}}
   
  +
\put(117,532){\sx{2.4}{\rot{87}$n\!=\!8$\ero}}
−
#include "ado.cin"
  
  +
\put(137,532){\sx{2.4}{\rot{86}$n\!=\!7$\ero}}
−
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
  
  +
\put(156,532){\sx{2.4}{\rot{85}$n\!=\!6$\ero}}
−
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
  
  +
\put(177,532){\sx{2.4}{\rot{84}$n\!=\!5$\ero}}
  +
\put(204,532){\sx{2.4}{\rot{83}$n\!=\!4$\ero}}
  +
\put(238,532){\sx{2.4}{\rot{82}$n\!=\!3$\ero}}
  +
\put(278,532){\sx{2.4}{\rot{79}$n\!=\!2$\ero}}
  +
\put(366,532){\sx{2.4}{\rot{71}$n\!=\!1$\ero}}
  +
%\put(264,350){\sx{3.1}{\rot{61}$y\!=\!x\!+\!\mathrm e^x$\ero}}
  +
%\put(427,528){\sx{2.3}{\rot{62}$c\!=\!0.6$\ero}}
  +
\put(448,528){\sx{2.4}{\rot{60}$n\!=\!0.4$\ero}}
  +
\put(469,530){\sx{2.4}{\rot{57}$n\!=\!0.3$\ero}}
  +
\put(491,532){\sx{2.4}{\rot{53}$n\!=\!0.2$\ero}}
  +
\put(517,534){\sx{2.4}{\rot{49}$n\!=\!0.1$\ero}}
   
  +
\put(541,531){\sx{2.4}{\rot{45}$n\!=\!0$\ero}}
−
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("TraIt.eps","w"); ado(o,604,604);
  
  +
% <br>
−
fprintf(o,"302 302 translate\n 100 100 scale\n");
  
  +
\put(521,462){\sx{2.4}{\rot{37}$n\!=\!-0.2$\ero}}
−
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  
  +
\put(517,423){\sx{2.4}{\rot{30}$n\!=\!-0.4$\ero}}
−
for(n=-3;n<4;n++) {M(-3,n)L(3,n)}
  
  +
\put(527,349){\sx{2.4}{\rot{17}$n\!=\!-1$\ero}}
−
for(m=-3;m<4;m++) {M(m,-3)L(m,3)}
  
  +
\put(524,261){\sx{2.4}{\rot{8}$n\!=\!-2$\ero}}
−
// M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)
 
  +
\put(523,222){\sx{2.4}{\rot{6}$n\!=\!-3$\ero}}
−
fprintf(o,".004 W S\n");
  
  +
\put(523,188){\sx{2.4}{\rot{4}$n\!=\!-4$\ero}}
−
// 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 0 RGB S\n");
  
  +
\put(523,161){\sx{2.4}{\rot{3}$n\!=\!-5$\ero}}
−
// 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 .5 0 RGB S\n");
  
  +
\put(523,141){\sx{2.4}{\rot{2}$n\!=\!-6$\ero}}
−
// M(0,0) L(12.03,12.03) fprintf(o,".04 W 0 1 0 RGB S\n");
  
  +
\put(523,122){\sx{2.4}{\rot{1}$n\!=\!-7$\ero}}
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(x)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
  +
\put(523,104){\sx{2.4}{\rot{1}$n\!=\!-8$\ero}}
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(x))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
  +
\end{picture}
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(x)))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
  +
\end{document}
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(x))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
  +
</pre>
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(x)))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
  +
==References==
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(x))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(tra(x)))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
−
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(tra(tra(x)))))))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
  
−
fprintf(o,".022 W 0 1 0 RGB S\n");
  
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(y)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(y))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(y)))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(y))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(y)))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
  
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(y))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
 
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(tra(y)))))))); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
 
−
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(tra(tra(y)))))))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
 
−
fprintf(o,".022 W 1 0 1 RGB S\n");
  
−
for(n=-20;n<21;n+=1){
  
−
DO(m,700){x=-3.01 +.02*m; y=Re(G(x)); y=Re(F(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
 
−
fprintf(o,".01 W 0 0 0 RGB S\n");
  
−
}
  
−
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o);
  
−
system("epstopdf TraIt.eps");
 
−
system( "open TraIt.pdf"); //for macintosh
  
−
getchar(); system("killall Preview"); // For macintosh
  
−
}
  
   
  +
{{ref}}
   
  +
{{fer}}
−
==[[Latex]] Generator of labels==
  
   
  +
==Keywords==
−
%<nowiki> %<br>
  
  +
«[[Abelfunction]]»,
−
% file PowPlo.pdf should be generated with the code above in order to compile the Latex document below. %<br>
  
  +
«[[AuTra]]»,
−
% Copyleft 2012 by Dmitrii Kouznetsov <br> %
  
  +
«[[Book]]»,
−
\documentclass[12pt]{article} % <br>
 
  +
«[[Entire Function with Logarithmic Asymptotic]]»,
−
\usepackage{geometry} % <br>
 
  +
«[[Iterate]]»,
−
\usepackage{graphicx} % <br>
 
  +
«[[Superfunction]]»,
−
\usepackage{rotating} % <br>
 
  +
«[[Superfunctions]]»,
−
\paperwidth 606pt % <br>
 
  +
«[[SuTra]]»,
−
\paperheight 606pt % <br>
 
  +
«[[Trappmann function]]»,
−
\topmargin -105pt % <br>
 
−
\oddsidemargin -73pt % <br>
 
−
\textwidth 1100pt % <br>
 
−
\textheight 1100pt % <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}(602,602) % <br>
 
−
%\put(10,10){\ing{PowPlo}} % <br>
 
−
\put(0,0){\ing{TraIt}} % <br>
 
−
\put(311,590){\sx{2.5}{$y$}} % <br>
 
−
\put(311,495){\sx{2.4}{$3$}} % <br>
 
−
\put(311,395){\sx{2.4}{$1$}} % <br>
 
−
\put(311,295){\sx{2.4}{$0$}} % <br>
 
−
\put(307,194){\sx{2.4}{$-1$}} % <br>
 
−
\put(307,093.4){\sx{2.4}{$-2$}} % <br>
 
−
% <br>
 
−
\put(083,308){\sx{2.4}{$-2$}} % <br>
 
−
\put(183,308){\sx{2.4}{$-1$}} % <br>
 
−
\put(297,308){\sx{2.4}{$0$}} % <br>
 
−
\put(397,308){\sx{2.4}{$1$}} % <br>
 
−
\put(497,308){\sx{2.4}{$2$}} % <br>
 
−
\put(590,308){\sx{2.5}{$x$}} % <br>
 
−
% <br>
 
−
\put(117,532){\sx{2.4}{\rot{87}$n\!=\!8$\ero}} % <br>
 
−
\put(137,532){\sx{2.4}{\rot{86}$n\!=\!7$\ero}} % <br>
 
−
\put(156,532){\sx{2.4}{\rot{85}$n\!=\!6$\ero}} % <br>
 
−
\put(177,532){\sx{2.4}{\rot{84}$n\!=\!5$\ero}} % <br>
 
−
\put(204,532){\sx{2.4}{\rot{83}$n\!=\!4$\ero}} % <br>
 
−
\put(238,532){\sx{2.4}{\rot{82}$n\!=\!3$\ero}} % <br>
 
−
\put(278,532){\sx{2.4}{\rot{79}$n\!=\!2$\ero}} % <br>
 
−
\put(366,532){\sx{2.4}{\rot{71}$n\!=\!1$\ero}} % <br>
 
−
%\put(264,350){\sx{3.1}{\rot{61}$y\!=\!x\!+\!\mathrm e^x$\ero}} % <br>
 
−
%\put(427,528){\sx{2.3}{\rot{62}$c\!=\!0.6$\ero}} % <br>
 
−
\put(448,528){\sx{2.4}{\rot{60}$n\!=\!0.4$\ero}} % <br>
 
−
\put(469,530){\sx{2.4}{\rot{57}$n\!=\!0.3$\ero}} % <br>
 
−
\put(491,532){\sx{2.4}{\rot{53}$n\!=\!0.2$\ero}} % <br>
 
−
\put(517,534){\sx{2.4}{\rot{49}$n\!=\!0.1$\ero}} % <br>
 
−
% <br>
 
−
\put(541,531){\sx{2.4}{\rot{45}$n\!=\!0$\ero}} % <br>
 
−
% <br>
 
−
\put(521,462){\sx{2.4}{\rot{37}$n\!=\!-0.2$\ero}} % <br>
 
−
\put(517,423){\sx{2.4}{\rot{30}$n\!=\!-0.4$\ero}} % <br>
 
−
\put(527,349){\sx{2.4}{\rot{17}$n\!=\!-1$\ero}} % <br>
 
−
\put(524,261){\sx{2.4}{\rot{8}$n\!=\!-2$\ero}} % <br>
 
−
\put(523,222){\sx{2.4}{\rot{6}$n\!=\!-3$\ero}} % <br>
 
−
\put(523,188){\sx{2.4}{\rot{4}$n\!=\!-4$\ero}} % <br>
 
−
\put(523,161){\sx{2.4}{\rot{3}$n\!=\!-5$\ero}} % <br>
 
−
\put(523,141){\sx{2.4}{\rot{2}$n\!=\!-6$\ero}} % <br>
 
−
\put(523,122){\sx{2.4}{\rot{1}$n\!=\!-7$\ero}} % <br>
 
−
\put(523,104){\sx{2.4}{\rot{1}$n\!=\!-8$\ero}} % <br>
 
−
\end{picture} % <br>
 
−
\end{document} % <br>
 
−
%</nowiki>
  
   
  +
«[[Суперфункции]]»
−
[[Category:Trappmann function]]
  
  +
−
[[Category:Iteration]]
  
−
[[Category:Superfunction]]
 +
[[Category:AuTra]]
−
[[Category:Abel function]]
 +
[[Category:Book]]
−
[[Category:Transfer function]]
 +
[[Category:BookPlot]]
 
[[Category:Explicit plot]]
  
[[Category:Explicit plot]]
−
[[Category:C++]]
 +
[[Category:Iterate]]
−
[[Category:Latex]]
 +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]
  +
[[Category:SuTra]]
  +
[[Category:Trappmann function]]

Latest revision as of 21:40, 12 January 2026


Fig.20.12 from page 288 of book «Superfunctions»[1], 2020.

The same picture is used also as Рис.20.12 at page 298 of the Russian version «Суперфункции» [2], 2014.

And also as Figure 5 at page 6537 of article Entire Function with Logarithmic Asymptotic [3] at Applied Mathematical Sciences, 2013.

The image shows the iterates of the Trappmann function \( \mathrm{tra} = z \mapsto z+\exp(z) \): \( y=\mathrm{tra}^n(y) \) versus \(x\) for various values of number \(n\) of iterate.

The \(n\)th iterate of the Trappmann function tra is expressed through its superfunction SuTra and the Abelfunction AuTra: \[\mathrm{tra}^n(z)= \mathrm{SuTra}(n+\mathrm{AuTra}(z)) \] The number \(n\) of iterate in this expression has no need to be integer.

Implementation

Namely this picture is generated using the representation of functions SuTra and AuTra through the Doya function, LambertW function] (Tania function), SuZex function.
So, files doya.cin LambertW.cin SuZex.cin and AuZex.cin should be loaded in order to compile the code below.

C++

#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 std::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 "doya.cin" // need for LambertW
#include "LambertW.cin" // need for AuZex
#include "SuZex.cin"
#include "AuZex.cin"
z_type tra(z_type z){ return exp(z)+z;}
z_type F(z_type z){ return log(suzex(z));}
z_type G(z_type z){ return auzex(exp(z));}
#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);
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;  FILE *o;o=fopen("trait.eps","w");  ado(o,604,604);
fprintf(o,"302 302 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(n=-3;n<4;n++) {M(-3,n)L(3,n)}
for(m=-3;m<4;m++) {M(m,-3)L(m,3)}
// M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)  
fprintf(o,".004 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 0 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 .5 0 RGB S\n");
// M(0,0) L(12.03,12.03) fprintf(o,".04 W 0 1 0 RGB S\n");
DO(m,700){x=-3.02+.02*m; y=Re(tra(x));                                   if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(x)));                              if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(x))));                         if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(x)))));                    if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(x))))));               if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(x)))))));          if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(tra(x))))))));     if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
DO(m,700){x=-3.02+.02*m; y=Re(tra(tra(tra(tra(tra(tra(tra(tra(x)))))))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;}
fprintf(o,".022 W 0 1 0 RGB S\n");
DO(m,700){y=-3.02+.02*m; x=Re(tra(y));                                   if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(y)));                              if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(y))));                         if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(y)))));                    if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(y))))));               if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(y)))))));          if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;} 
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(tra(y))))))));     if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;} 
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(tra(tra(tra(tra(tra(y)))))))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;} 
fprintf(o,".022 W 1 0 1 RGB S\n");
M(-3,-3)L(3,3) fprintf(o,".022 W 0 .6 1 RGB S\n");
for(n=-20;n<21;n+=1){
  DO(m,700){x=-3.01 +.02*m; y=Re(G(x)); y=Re(F(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}  
  fprintf(o,".01 W 0 0 0 RGB S\n");
                   }
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
      system("epstopdf trait.eps"); 
      system(    "open trait.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 606pt
\paperheight 606pt
\topmargin -105pt
\oddsidemargin -73pt
\textwidth 1100pt
\textheight 1100pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt
\pagestyle{empty}
\begin{document}
\begin{picture}(602,602)
%\put(10,10){\ing{PowPlo}}
\put(0,0){\ing{TraIt}}
\put(311,590){\sx{2.5}{$y$}}
\put(311,495){\sx{2.4}{$2$}}
\put(311,395){\sx{2.4}{$1$}}
\put(311,295){\sx{2.4}{$0$}}
\put(307,194){\sx{2.4}{$-1$}}
\put(307,093.4){\sx{2.4}{$-2$}}

\put(083,308){\sx{2.4}{$-2$}}
\put(183,308){\sx{2.4}{$-1$}}
\put(297,308){\sx{2.4}{$0$}}
\put(397,308){\sx{2.4}{$1$}}
\put(497,308){\sx{2.4}{$2$}}
\put(590,308){\sx{2.5}{$x$}}

\put(117,532){\sx{2.4}{\rot{87}$n\!=\!8$\ero}}
\put(137,532){\sx{2.4}{\rot{86}$n\!=\!7$\ero}}
\put(156,532){\sx{2.4}{\rot{85}$n\!=\!6$\ero}}
\put(177,532){\sx{2.4}{\rot{84}$n\!=\!5$\ero}}
\put(204,532){\sx{2.4}{\rot{83}$n\!=\!4$\ero}}
\put(238,532){\sx{2.4}{\rot{82}$n\!=\!3$\ero}}
\put(278,532){\sx{2.4}{\rot{79}$n\!=\!2$\ero}}
\put(366,532){\sx{2.4}{\rot{71}$n\!=\!1$\ero}}
%\put(264,350){\sx{3.1}{\rot{61}$y\!=\!x\!+\!\mathrm e^x$\ero}}
%\put(427,528){\sx{2.3}{\rot{62}$c\!=\!0.6$\ero}}
\put(448,528){\sx{2.4}{\rot{60}$n\!=\!0.4$\ero}}
\put(469,530){\sx{2.4}{\rot{57}$n\!=\!0.3$\ero}}
\put(491,532){\sx{2.4}{\rot{53}$n\!=\!0.2$\ero}}
\put(517,534){\sx{2.4}{\rot{49}$n\!=\!0.1$\ero}}

\put(541,531){\sx{2.4}{\rot{45}$n\!=\!0$\ero}}
% <br> 
\put(521,462){\sx{2.4}{\rot{37}$n\!=\!-0.2$\ero}}
\put(517,423){\sx{2.4}{\rot{30}$n\!=\!-0.4$\ero}}
\put(527,349){\sx{2.4}{\rot{17}$n\!=\!-1$\ero}}
\put(524,261){\sx{2.4}{\rot{8}$n\!=\!-2$\ero}}
\put(523,222){\sx{2.4}{\rot{6}$n\!=\!-3$\ero}}
\put(523,188){\sx{2.4}{\rot{4}$n\!=\!-4$\ero}}
\put(523,161){\sx{2.4}{\rot{3}$n\!=\!-5$\ero}}
\put(523,141){\sx{2.4}{\rot{2}$n\!=\!-6$\ero}}
\put(523,122){\sx{2.4}{\rot{1}$n\!=\!-7$\ero}}
\put(523,104){\sx{2.4}{\rot{1}$n\!=\!-8$\ero}}
\end{picture}
\end{document}

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