File:ArcShokaMapT.png

Complex map of the ArcShoka function:

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

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.

%