File:SuTraMapN.png

Complex map of the Entire Function with Logarithmic Asymptotic.

This function is defined thought the SuTra function ^[1][2][3]:

\(\Phi(z) = - \ \)SuTra\( (-z) \)

It is shown assuming \(u+\mathrm i v= \Phi(x+\mathrm i y) \) in the \(x,y\) plane with lines \(u=\rm const\) and lines \(v=\rm const\) .

Here, SuTra is Superfunction of the Trappmann function \(\mathrm{tra} = z \mapsto z+\exp(z) \ \).

The Trappmann function has no fixed point. For this reason, it had been considered to be especially interesting in the role of the transfer function: until year 2013, its superfunction had been believed to be very difficult to construct, if at all.

Mnemonic for the filename: function SuTra is Mapped with Negative of its values and Negative of the argument.

C++ generator of the map

/* Files ado.cin, conto.cin, Tania.cin, LambertW.cin, SuZex.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 "tania.cin"
 #include "LambertW.cin"
 #include "SuZex.cin"
 //#include "AuZex.cin" 

 z_type tra(z_type z) {return z+exp(z);}

// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
 z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
                                        return tra(sutra(z-1.));}

int 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();
 int M=1001,M1=M+1;
 int N=1001,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("SuTraMapM.eps","w");  ado(o,2002,2002);
 fprintf(o,"1001 1001 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 DO(m,M1) X[m]=-10+.02*(m-.5);
 DO(n,N1) Y[n]=-10+.02*(n-.5); 
 //for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
 for(m=-10;m<11;m++){M(m,-10) L(m,10)  }
 for(n=-10;n<11;n++){M(  -10,n) L(10,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); //if(abs(z+2.)>.019)
 // c=AuZex01(z-1.);
 // c=AuZexAsy(LambertW(z))+1.;
 //c=log(suzex(z));
   c=-sutra(-z);
 // p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
  p=Re(c); q=Im(c); if(p>-19 && p<19 && ( x<2. ||  fabs(q)>1.e-12 && fabs(p)>1.e-12) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
        }}
 fprintf(o,"1 setlinejoin 1 setlinecap\n");
  p=1.2;q=.4;
 /*  p=9;q=.16;
  conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".02 W 1 0 1 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".01 W .2 .2 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(13.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(12.  ),-p,p); fprintf(o,".03 W 0 0 1 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(11.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N,(10.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (9.  ),-p,p); fprintf(o,".03 W 0 1 1 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (8.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (7.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (6.  ),-p,p); fprintf(o,".04 W 0 .5 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (5.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (4.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (3.  ),-p,p); fprintf(o,".02 W 1 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (2.  ),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  conto(o,g,w,v,X,Y,M,N, (1.  ),-p,p); fprintf(o,".02 W .5 0 0 RGB S\n");
 */
 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.  ),-2*p,2*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,"0 setlinejoin 0 setlinecap\n");
 // M(-10,0)L(0,0) fprintf(o,"1 1 1 RGB .02 W S\n");
 //#include "plofu.cin"
 fprintf(o,"showpage\n");
 fprintf(o,"%c%cTrailer\n",'%','%');
 fclose(o);  free(f); free(g); free(w);
       system("epstopdf SuTraMapM.eps"); 
       system(    "open SuTraMapM.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 return 0;
 }

Latex generator of the labels

\documentclass[12pt]{article} % <br>
\paperwidth 2010px
\paperheight 2008px
\textwidth 2394px
\textheight 2300px
\topmargin -100px
\oddsidemargin -70px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\parindent 0pt
\pagestyle{empty}
\begin{document}
\parindent 0pt
\sx{1}{\begin{picture}(2002,2002) 
\put(30,1950){\sx{8}{$y$}}
\put(30,1780){\sx{7}{$8$}}
\put(30,1580){\sx{7}{$6$}}
\put(30,1380){\sx{7}{$4$}}
\put(30,1180){\sx{7}{$2$}}
\put(30,0980){\sx{7}{$0$}}
\put(-22,780){\sx{7}{$-2$}}
\put(-22,580){\sx{7}{$-4$}}
\put(-22,380){\sx{7}{$-6$}}
\put(-22,180){\sx{7}{$-8$}}
\put(120, 6){\sx{7}{$-8$}}
\put(320, 6){\sx{7}{$-6$}}
\put(520, 6){\sx{7}{$-4$}}
\put(720, 6){\sx{7}{$-2$}}
\put(988, 6){\sx{7}{$0$}}
\put(1188, 6){\sx{7}{$2$}}
\put(1388, 6){\sx{7}{$4$}}
\put(1588, 6){\sx{7}{$6$}}
\put(1788, 6){\sx{7}{$8$}}
\put(1958, 6){\sx{8}{$x$}}
\put(0,0){\ing{SuTraMapM}}
%\zoomax
\put(180,1480){\sx{7}{\rot{56}$u\!=\!2.2$\ero}}
\put(322,1430){\sx{7}{\rot{56}$u\!=\!2$\ero}}
\put(480,1238){\sx{7}{\rot{58}$u\!=\!1.6$\ero}}

\put(1518,1238){\sx{7}{\rot{16}$v\!=\!0.4$\ero}}
\put(1518,1104){\sx{7}{\rot{8}$v\!=\!0.2$\ero}}

\put(1350,1590){\sx{7}{\rot{50}$v\!=\!1$\ero}}
\put(710,1632){\sx{7}{\rot{-68}$v\!=\!2$\ero}}
\put(310,0984){\sx{7}{$v\!=\!0$}} \put(1510,0984){\sx{7}{$v\!=\!0$}}

\put(1244,0920){\sx{7}{\rot{90}$u\!=\!1$\ero}}
\put(1762,0920){\sx{7}{\rot{90}$u\!=\!2$\ero}}

\put(1490,876){\sx{7}{\rot{-11}$v\!=\!-0.2$\ero}}
\put(1490,738){\sx{7}{\rot{-19}$v\!=\!-0.4$\ero}}
\put(1480,610){\sx{7}{\rot{-32}$v\!=\!-0.6$\ero}}
\end{picture}}
\end{document}
%

Warning

Due to the permanent attacks by KGB (Roskomnadzor, «Фабрика троллей» ^[4]) on servers of cluster "Mizugadro", the codes (generators) of pictures run at different computers with different operational systems. Historically, some of them are not case-sensitive. This caused confusions with UPPERCASE/lowercase versions of the filenames.
Editor tries to eliminate the confusion, but some manual capitalization of some of the included files still may be required.
Editor hopes the problem with the attacks to be resolved after Demilitarization and denazification (Демилитаризация и денацификация) of Roskomnadzor (Роскомнадзор) and Lubyanka (Лубянка).
While KGB and Roskomnadzor attack Mizugadro, the response of the server may be slow. The copying of the content to other servers is encouraged - especially if you find some articles and/or pictures at mizugadro valuable, useful, important. In this way, «manuscripts don't burn.»^[5].

References

Keywords

«ado.cin», «conto.cin», «Complex map», «Tania.cin», «LambertW.cin», «SuZex.cin», «Entire Function with Logarithmic Asymptotic», «Logarithm», «Superfunctions», «SuTra», «Trappmann function»,