Difference between revisions of "File:AuTraMapT.jpg"

Fig.20.11 at page 287 of book «Superfunctions»^[1], 2020.

It appears also as Рис.20.11 at page 297 of the Russian version «Суперфункции»^[2], 2014.

The picture shows the complex map of function AuTra. It is Abelfunction for the elementary Trappmann function \(\mathrm{tra} = z \mapsto z+\exp(z)\).

AuTra is inverse function of SuTra. SuTra is Entire Function with Logarithmic Asymptotic, it is described also in the Applied Mathematical Sciences ^[3], 2013.

Complex map is shown with \(u\!+\!i v\!=\! \mathrm{AuTra}(x\!+\!\mathrm i y)\).

C++ generator of curves

// Files ado.cin, conto.cin, arctran.cin, sutran.cin autran.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 "conto.cin"

#include "arctran.cin"
#include "sutran.cin"
#include "autran.cin"
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=501,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("AuTraMap.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+.04*(m-.5);
DO(n,N1) Y[n]=-10+.04*(n-.49); 
//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]=9999;
                       f[m*N1+n]=9999;}
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=auzex(z);
//c=suzex(c);
  c=autran(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<1000 && p>-1000 && q<1000 && q>-1000 
// ( 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=4.;q=.6;
for(m=-6;m<6;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".008 W 0 .6 0 RGB S\n");
for(m=0;m<6;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".008 W .9 0 0 RGB S\n");
for(m=0;m<6;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".008 W 0 0 .9 RGB S\n");

for(m= 1;m<13;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".032 W .8 0 0 RGB S\n");
for(m= 1;m<13;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".032 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-4*p,4*p); fprintf(o,".032 W .5 0 .5 RGB S\n");
for(m=-12;m<13;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");


fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(-10, M_PI)L(-1, M_PI) 
M(-10,-M_PI)L(-1,-M_PI) fprintf(o,"1 1 1 RGB .08 W S\n");
DO(n,26){M(-1.-.4*n, M_PI)L(-1.-.4*(n+.4), M_PI) }
DO(n,26){M(-1.-.4*n,-M_PI)L(-1.-.4*(n+.4),-M_PI) }   fprintf(o,"0 0 0 RGB .12 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 AuTraMap.eps"); 
      system(    "open AuTraMap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
return 0;
}

Latex generator of labels

%File AuTraMap.pdf should be generated with the code above in order to compile the Latex document below.

\documentclass[12pt]{article}
\paperwidth 2100px
\paperheight 2072px
\textwidth 2394px
\textheight 2300px
\topmargin -101px
\oddsidemargin -80px
\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}
\newcommand \zoomax {
\put(40,2048){\sx{8}{$y$}}
\put(40,1870){\sx{7}{$8$}}
\put(40,1670){\sx{7}{$6$}}
\put(40,1470){\sx{7}{$4$}}
\put(40,1270){\sx{7}{$2$}}
\put(40,1070){\sx{7}{$0$}}
\put(-12,870){\sx{7}{$-2$}}
\put(-12,670){\sx{7}{$-4$}}
\put(-12,470){\sx{7}{$-6$}}
\put(-12,270){\sx{7}{$-8$}}
\put(002, 28){\sx{7}{$-\!10$}}
\put(220, 28){\sx{7}{$-8$}}
\put(420, 28){\sx{7}{$-6$}}
\put(620, 28){\sx{7}{$-4$}}
\put(820, 28){\sx{7}{$-2$}}
\put(1088, 28){\sx{7}{$0$}}
\put(1288, 28){\sx{7}{$2$}}
\put(1488, 28){\sx{7}{$4$}}
\put(1688, 28){\sx{7}{$6$}}
\put(1888, 28){\sx{7}{$8$}}
\put(2058, 28){\sx{7}{$x$}}
%\put(2166, 28){\sx{7}{$x$}}
}
\parindent 0pt
\sx{1}{\begin{picture}(2102,2094)
%\put(40,20){\ing{b271tMap3}}
%\put(40,20){\ing{ExpMap}}
\put(98,88){\ing{AuTraMap}}
\zoomax
\put(218,1384){\sx{9}{\bf cut}}
\put(218, 752){\sx{9}{\bf cut}}
%
\put( 224,1836){\sx{8}{\rot{35}$v\!=\!0.6$\ero}}
\put( 1590,1496){\sx{8}{\rot{70}$v\!=\!0.4$\ero}}
\put( 1800,1360){\sx{8}{\rot{38}$v\!=\!0.2$\ero}}
\put( 140,1068){\sx{9}{$v\!=\!0$}}
\put(1874,1068){\sx{9}{$v\!=\!0$}}
\put( 1740, 814){\sx{8}{\rot{-40}$v\!=\!-0.2$\ero}}
\put( 194, 306){\sx{8}{\rot{-35}$v\!=\!-0.6$\ero}}
%
\put( 606, 1920){\sx{9}{\rot{-64}$u\!=\!3$\ero}}
\put( 1500, 992){\sx{8.2}{\rot{90}$u\!=\!2$\ero}}
\put( 1630, 960){\sx{8}{\rot{90}$u\!=\!2.2$\ero}}
\put( 1846, 960){\sx{8}{\rot{90}$u\!=\!2.4$\ero}}
\put( 1724, 400){\sx{8}{\rot{22}$u\!=\!2.6$\ero}}
\put( 670, 290){\sx{9}{\rot{60}$u\!=\!3$\ero}}
%
\end{picture}}
\end{document}

References

Keywords

«Abel function», «Abelfunction», «AuTra», «Complex map», «Inverse function», «Superfunctions», «Trappmann function»,