File:ArcShokaMapT.png

Complex map of the ArcShoka function:

\[ u\! + \! \mathrm i v = \mathrm{ArcShoka}(x \!+\! \mathrm i y)\]

This map is used as Fig.5.13 at page 58 of book «Superfunctions» ^[1][2].

C++ Generator of curces

//Files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ 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"
 
 z_type Shoko(z_type z){ return log(1.+exp(z)*(M_E-1.)); }
 z_type Shoka(z_type  z) { return z + log(exp(-z)+(M_E-1.)); }
 z_type ArcShoka(z_type z){  return  z + log((1.-exp(-z))/(M_E-1.)) ;}
 // z_type ArcShoka(z_type z){ DB x,y,p; int n; 
 // y=Im(z); n=round(y/(2.*M_PI)); p=n*(2*M_PI); z-=z_type(0,p); 
 //   return log((exp(z)-1.)/(M_E-1.)) + z_type(0,p); }

 main(){ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d;
 r=log(1./(M_E-1.)); printf("r=%16.14f\n",r); 
 int M=400,M1=M+1;
 int N=800,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("ArcShokaMap.eps","w");ado(o,162,162);
 fprintf(o,"81 81 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-8.01+.0401*(m);
 DO(n,N1)Y[n]=-8.01+.0201*(n+.5);
 for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
 for(n=-8;n<9;n++){     M(  -8,n)L(8,n)}
 fprintf(o,".008 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]; //printf("%5.2f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
 // c=Tania(z); p=Re(c);q=Im(c);  
 // c=Shoko(z); p=Re(c);q=Im(c);  
 c=ArcShoka(z); 
 // c=Shoka(c); 
 p=Re(c);q=Im(c);  
 if(p>-99. && p<99. &&  q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
        }}
 fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=.6;q=.5;
 for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .6 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .9 0 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .9 RGB S\n");
 for(m=1;m<10;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n");
 for(m=1;m<10;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n");
 for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
 for(y=-2*M_PI;y<7.;y+=2*M_PI) {
     M(0,y)L(-8.1,y) fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n");
    for(m=0;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 1 .5 0 RGB S\n");
    for(m=2;m<81;m+=4) {x=-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 0 .5 1 RGB S\n");
                           }
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf ArcShokaMap.eps");    
       system(    "open ArcShokaMap.pdf");
 printf("r=%16.14f  %16.14f\n",r,sqrt(M_PI*M_PI+r*r)); 
       getchar(); system("killall Preview");
 }
//

Latex generator fo labels

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

 % Copyleft 2011 by Dmitrii Kouznetsov %

 \documentclass[12pt]{article} %

 \usepackage{geometry} %

 \usepackage{graphicx} %

 \usepackage{rotating} %

 \paperwidth 854pt %

 \paperheight 844pt %

 \topmargin -96pt %

 \oddsidemargin -98pt %

 \textwidth 1100pt %

 \textheight 1100pt %

 \pagestyle {empty} %

 \newcommand \sx {\scalebox} %

 \newcommand \rot {\begin{rotate}} %<br>
 \newcommand \ero {\end{rotate}} %

 \newcommand \ing {\includegraphics} %

 \begin{document} %<br>
 \sx{5}{ \begin{picture}(164,165) %<br>
 \put(6,5){\ing{ArcShokaMap}} %<br>
 \put(2,163){\sx{.7}{$y$}} %<br>
 \put(2,144){\sx{.6}{$6$}} %<br>
 \put(2,124){\sx{.6}{$4$}} %<br>
 \put(2,104){\sx{.6}{$2$}} %<br>
 % \put(3,116){\sx{.6}{$\pi$}} %<br> 
 \put(18,147){\sx{.8}{\bf cut}} %<br>
 \put(2, 84){\sx{.6}{$0$}} %<br>
 \put(18,84){\sx{.8}{\bf cut}} %<br>
 \put(-2,64){\sx{.6}{$-2$}} %<br>
 \put(18,20.8){\sx{.8}{\bf cut}} %<br>
 \put(-2,44){\sx{.6}{$-4$}} %<br>
 \put(-2,24){\sx{.6}{$-6$}} %<br>
 \put( 22,0){\sx{.6}{$-6$}} %<br>
 \put( 42,0){\sx{.6}{$-4$}} %<br>
 \put( 62,0){\sx{.6}{$-2$}} %<br>
 \put( 86,0){\sx{.6}{$0$}} %<br>
 \put(106,0){\sx{.6}{$2$}} %<br>
 \put(126,0){\sx{.6}{$4$}} %<br>
 \put(146,0){\sx{.6}{$6$}} %<br>
 \put(164,0){\sx{.7}{$x$}} %<br>
 \put( 86.5, 108){\rot{90}\sx{.7}{$u\!=\!0$}\ero}%<br>
 \put(102.6, 107){\rot{90}\sx{.7}{$u\!=\!1$}\ero}%<br>
 \put(114.2, 107){\rot{90}\sx{.7}{$u\!=\!2$}\ero}%<br>
 \put(124.6, 107){\rot{90}\sx{.7}{$u\!=\!3$}\ero}%<br>
 \put(132,154.4){\sx{.8}{$v\!=\!7$}}%<br>
 \put(132,144.3){\sx{.8}{$v\!=\!6$}}%<br>
 \put(132,134.3){\sx{.8}{$v\!=\!5$}}%<br>
 \put(132,124.2){\sx{.8}{$v\!=\!4$}}%<br>
 \put(132,114.2){\sx{.8}{$v\!=\!3$}}%<br>
 \put(132,104.2){\sx{.8}{$v\!=\!2$}}%<br>
 \put(132,  94.2){\sx{.8}{$v\!=\!1$}}%<br>
 \put(132,  84){\sx{.8}{$v\!=\!0$}}%<br>
 \put(132,  73.9){\sx{.8}{$v\!=\!-\!1$}}%<br>
 \put(132,  63.4){\sx{.8}{$v\!=\!-\!2$}}%<br>
 \put(132,  53.4){\sx{.8}{$v\!=\!-\!3$}}%<br>
 \put(132,  43.4){\sx{.8}{$v\!=\!-\!4$}}%<br>
 \put(132,  33.4){\sx{.8}{$v\!=\!-\!5$}}%<br>
 \put(132,  23.4){\sx{.8}{$v\!=\!-\!6$}}%<br>
 \put(132,  13.4){\sx{.8}{$v\!=\!-\!7$}}%<br>
 \end{picture} %

 } %

\end{document}
%
References

Keywords