File:TraMapT.jpg

Fig.20.2 from page 269 of book «Superfunctions»^[1], 2020.

The image is used also as Рис.20.2 at page 278 of the Russian version «Суперфункции» ^[2], 2014.

The figure shows the complex map of the Trappmann function \(\mathrm{tra}(z)=z+\mathrm e^z\):

\( u\!+\!\mathrm i v\!=\!\mathrm{tra}(x\!+\!\mathrm i y) \)

The Trappmann function is interesting as a transfer function, because it has no fixed point.
The superfunction (SuTra) for this transfer function was believed to be difficult to construct if at al.
However, it happened to be not a case; the construction is published at Applied Mathematical Sciences ^[3] in 2013.
The construction of the superfunction SuTra, the abelfunction AuTra, iterates of the Trappmann function and their properties are described also in the books cited.

C++ generator of curves

/* Files ado.cin and conto.cin should be loaded 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 "conto.cin"
 //#include "fsexp.cin"
 //#include "fslog.cin"

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

 main(){ int j,k,m,n; DB x1,x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=601,M1=M+1;
 int N=601,N1=N+1;
 DB X[M1],Y[N1];
 DB *g, *f, *w; // w is working array.
 g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("TraMap.eps","w");  ado(o,1202,1202);
 fprintf(o,"601 601 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 DO(m,M1) X[m]=-6.+.02*(m-.5);
 DO(n,N1) Y[n]=-6.+.02*(n-.5); 
 //for(n=0;n<N1;n++) { Y[n]=0.6*sinh((3./200.)*(n-200.5)); printf("%3d %9.6f\n",n,Y[n]); }
 for(m=-6;m<7;m++) {M(m,-6)L(m,6)}
 for(n=-6;n<7;n++) {M(  -6,n)L(6,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                        f[m*N1+n]=999;}
 DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);  c=tra(z);  p=Re(c); q=Im(c);
 //        if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) 
 g[m*N1+n]=p;
 //        if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) 
 f[m*N1+n]=q;
        }}
 fprintf(o,"1 setlinejoin 1 setlinecap\n");  p=24.;q=.2;
 for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
 for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
 for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
 for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
 for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
 for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
 fprintf(o,"showpage\n");  fprintf(o,"%c%cTrailer\n",'%','%');
 fclose(o);  free(f); free(g); free(w);
       system("epstopdf TraMap.eps"); 
       system(    "open TraMap.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }

Latex generator of labels

\documentclass[12pt]{article}
\paperheight 1228px
\paperwidth 1236px
\textwidth 1394px
\textheight 1300px
\topmargin -104px
\oddsidemargin -78px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
\newcommand \zoomax {
\put(18,1206){\sx{3.3}{$y$}}
\put(18,1113){\sx{3}{$5$}}
\put(18,1013){\sx{3}{$4$}}
\put(18, 913){\sx{3}{$3$}}
\put(18, 813){\sx{3}{$2$}}
\put(18, 713){\sx{3}{$1$}}
\put(18, 613){\sx{3}{$0$}}
\put(-6, 513){\sx{3}{$-1$}}
\put(-6, 413){\sx{3}{$-2$}}
\put(-6, 313){\sx{3}{$-3$}}
\put(-6, 213){\sx{3}{$-4$}}
\put(-6, 113){\sx{3}{$-5$}}
\put(-6, 013){\sx{3}{$-6$}}
\put(014, -5){\sx{3}{$-6$}}
\put(114, -5){\sx{3}{$-5$}}
\put(214, -5){\sx{3}{$-4$}}
\put(314, -5){\sx{3}{$-3$}}
\put(414, -5){\sx{3}{$-2$}}
\put(514, -5){\sx{3}{$-1$}}
\put(635, -5){\sx{3}{$0$}}
\put(735, -5){\sx{3}{$1$}}
\put(835, -5){\sx{3}{$2$}}
\put(935, -5){\sx{3}{$3$}}
\put(1035, -5){\sx{3}{$4$}}
\put(1135, -5){\sx{3}{$5$}}
\put(1227,-4){\sx{3}{$x$}}
}
\parindent 0pt
\sx{1}{\begin{picture}(1252,1220)
%\put(40,20){\ing{b271tMap3}}
%\put(40,20){\ing{ExpMap}}
%\put(40,20){\ing{suzexD1map}}
\put(40,20){\ing{tramap}}
\zoomax
\put(90,1116){\sx{4}{$v\!=\!5$}}
\put(90,1015){\sx{4}{$v\!=\!4$}}
\put(90,914){\sx{4}{$v\!=\!3$}}
\put(90,813){\sx{4}{$v\!=\!2$}}
\put(90,712){\sx{4}{$v\!=\!1$}}
\put(90,611){\sx{4}{$v\!=\!0$}}
\put(90,510){\sx{4}{$v\!=\!-1$}}
\put(90,409){\sx{4}{$v\!=\!-2$}}
\put(90,308){\sx{4}{$v\!=\!-3$}}
\put(90,207){\sx{4}{$v\!=\!-4$}}
\put(90,106){\sx{4}{$v\!=\!-5$}}
\put(257,555){\sx{4}{\rot{90}$u\!=\!-4$\ero}}
\put(355,555){\sx{4}{\rot{90}$u\!=\!-3$\ero}}
\put(447,555){\sx{4}{\rot{90}$u\!=\!-2$\ero}}
\put(534,555){\sx{4}{\rot{90}$u\!=\!-1$\ero}}
\put(600,572){\sx{4}{\rot{90}$u\!=\!0$\ero}}
\put(656,574){\sx{4}{\rot{90}$u\!=\!1$\ero}}

\put(990,678){\sx{4}{\rot{-15}$v\!=\!16$\ero}}
\put(990,550){\sx{4}{\rot{9}$v\!=\!-16$\ero}}
\put(998,507){\sx{4}{\rot{-11}$u\!=\!16$\ero}}
\put(996,380){\sx{4}{\rot{13}$u\!=\!-16$\ero}}

\end{picture}}
\end{document}
%

References

Keywords

«Complex map», «Elementary function», «Superfunctions», «Transfer function», «Trappmann function»,