File:AsincmapT.png

Complex map of function ArcSinc.

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

C++ implementation as asinc.cin

The text below is supposed to be stored as asinc.cin

z_type acos(z_type z){
if(Im(z)<0){if(Re(z)>=0){return I*log( z + sqrt(z*z-1.) );} 
                   else{return I*log( z - sqrt(z*z-1.) );}}
           if(Re(z)>=0){return -I*log( z + sqrt(z*z-1.) );} 
                  else {return -I*log( z - sqrt(z*z-1.) );} }

z_type asin(z_type z){
if(Im(z)<0){if(Re(z)>=0){return M_PI/2.-I*log( z + sqrt(z*z-1.) );} 
                  else {return M_PI/2.-I*log( z - sqrt(z*z-1.) );}}
          if(Re(z)>=0){return M_PI/2.+I*log( z + sqrt(z*z-1.) );} 
                 else {return M_PI/2.+I*log( z - sqrt(z*z-1.) );} }

z_type sinc(z_type z){ DB x=Re(z); DB y=Im(z); if(x*x+y*y>.01) return sin(z)/z;
z*=z; return 1.+z*(-1./6.+z*(1./120.+z*(-1./5040+z*(1./362880.)))); }

z_type sincp(z_type z){ DB x=Re(z); DB y=Im(z); z_type c; 
if(x*x+y*y>.01) return (cos(z)-sinc(z))/z;
c=z*z; return z*(-1./3.+c*(1./30.+c*(-1./840.+c*(1./45630+c*(-1./399168.))))); }

DB Namihei=4.493409457909062;
DB Katsuo=-0.21723362821122164;
z_type asincL(z_type z){int n; z_type s=Namihei-sqrt((-2./Katsuo)*(z-Katsuo));
DO(n,6)s+=(z-sinc(s))/sincp(s);  return s;}

z_type asincU(z_type z){int n; z_type s=1.4*acos(z);
  DO(n,6)s+=(z-sinc(s))/sincp(s);  return s;}

z_type asinc(z_type z){DB x=Re(z),y=fabs(Im(z));
    if(y<1. && x<0. && x>-2.6 ) return asincL(z);
                                return asincU(z); }

C++ generator of curves

// File asinc.cin above, and also files ado.cin and conto.cin

// sould be loaded in the working directory in order to compile the C++ code below.

#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 "asinc.cin"

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=801,M1=M+1;
int N=401,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("asincmap.eps","w");ado(o,82,82);
fprintf(o,"41 41 translate\n 10 10 scale\n"); 
DO(m,400) X[m]=-4.+.01*m;
     X[400]=-.002;
     X[401]= .002;
for(m=402;m<M1;m++) X[m]=-4.+.01*(m-1.);
DO(n,200)Y[n]=-4.+.02*n;
      Y[200]=-.002;
      Y[201]= .002;
for(n=202;n<N1;n++) Y[n]=-4.+.02*(n-1.);
for(m=-4;m<5;m++){if(m==0){M(m,-4.1)L(m,4.1)} else{M(m,-4)L(m,4)}}
for(n=-4;n<5;n++){     M(  -4,n)L(4,n)}
fprintf(o,".01 W 0 0 0 RGB S\n");
M(Katsuo,-4)L(Katsuo,4);
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]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
      c=asinc(z);
 //      c=sinc(c);
 //      p=-log(abs(c-z)/abs(c+z))/log(10.);
   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;
}
                     }}
// #include "plodi.cin"
/*
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=22;
conto(o,g,w,v,X,Y,M,N, (15.4 ),-p,p); fprintf(o,".02 W 0 .6 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.  ),-p,p); fprintf(o,".01 W 0 0 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,".02 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,".02 W 0 1 0 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,".02 W 1 0 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,".03 W 0 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 .6 0 0 RGB S\n");
*/
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.5;q=.5;
for(m=-11;m<11;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<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,".007 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,".007 W 0 0 .9 RGB S\n");
for(m=1;m<11;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".02 W .9 0 0 RGB S\n");
for(m=1;m<11;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 .9 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".02 W .6 0 .6 RGB S\n");
for(m=-9;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
    m=0;          conto(o,g,w,v,X,Y,M,N, (0.+m),-2*p,2*p); fprintf(o,".02 W 0 0 0 RGB S\n");
for(m=1;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");
 //#include "plofu.cin"
M(-4.01,0)L(Katsuo,0)M(1,0)L(4.01,0)  fprintf(o,"0 setlinecap .021 W 1 1 1 RGB S\n");
DO(n,39){M(Katsuo-.1*n,0) L(Katsuo-.1*(n+.5),0)}
DO(n,31){M(1+.1*n,0) L(1+.1*(n+.5),0)}
fprintf(o,"0 setlinecap .024 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
     system("epstopdf asincmap.eps");
     system(    "open asincmap.pdf");            
     getchar(); system("killall Preview");//for mac
}

Latex generator of labels

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

% \documentclass[12pt]{article} %<br> \paperheight 838px %<br> \paperwidth 844px %<br> \textwidth 1294px %<br> \textheight 1200px %<br> \topmargin -80px %<br> \oddsidemargin -80px %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \newcommand \rmi {\mathrm{i}} %<br> \begin{document} %<br> \newcommand \zoomax { %<br> \put(16,820){\sx{4.4}{$y$}} %<br> \put(16,630){\sx{4}{$2$}} %<br> \put(16,430){\sx{4}{$0$}} %<br> \put(-4, 230){\sx{4}{$-\!2$}} %<br> \put(220, 5){\sx{4}{$-\!2$}} %<br> \put(443, 5){\sx{4}{$0$}} %<br> \put(643, 5){\sx{4}{$2$}} %<br> \put(831,6){\sx{4}{$x$}} %<br> } %<br> \parindent 0pt %<br> %\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} %<br> \begin{picture}(816,816) %<br> \put(40,30){\sx{10}{\ing{asincmap}}} %<br> \zoomax %<br> \put(100,548){\sx{4}{\rot{-11}$u\!=\!3.6$\ero}} %<br> \put(90,484){\sx{4}{\rot{-1}$u\!=\!3.8$\ero}} %<br> \put(90,375){\sx{4}{\rot{1}$u\!=\!3.8$\ero}} %<br> \put(209,150){\sx{4}{\rot{40}$u\!=\!3$\ero}} %<br> \put(501, 40){\sx{4}{\rot{86}$u\!=\!2$\ero}} %<br> %<br> \put(654,634){\sx{4}{\rot{56}$u\!=\!1.2$\ero}} %<br> \put(686,610){\sx{4}{\rot{47}$u\!=\!1$\ero}} %<br> \put(698,574){\sx{4}{\rot{36}$u\!=\!0.8$\ero}} %<br> \put(720,540){\sx{4}{\rot{26}$u\!=\!0.6$\ero}} %<br> \put(722,500){\sx{4}{\rot{20}$u\!=\!0.4$\ero}} %<br> \put(720,463){\sx{4}{\rot{10}$u\!=\!0.2$\ero}} %<br> %<br> \put(70,432){\sx{4}{\bf cut}} %<br> \put(690,432){\sx{4}{\bf cut}} %<br> \put(436,703){\sx{4}{$v\!=\!-3$}} %<br> \put(432,550){\sx{4}{$v\!=\!-2$}} %<br> \put(441,430){\sx{4}{$v\!=\!0$}} %<br> \put(450,385){\sx{4}{$v\!=\!1$}} %<br> \put(452,307){\sx{4}{$v\!=\!2$}} %<br> \put(452,153){\sx{4}{$v\!=\!3$}} %<br> \end{picture} %<br> \end{document} %<br> % %