Difference between revisions of "File:Ausinmap52r5.jpg"

Figure 12.4 from page 154 of book «Superfunctions» ^[1]:

Complex map of function AuSin:

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

C++ generator of curves

Files ado.cin, arcsin.cin, susin.cin, conto.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"sutran.cin"

//z_type arcsin(z_type z){ return -I* log(I*z + sqrt(1.-z*z) ); }

z_type arcsin(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.) );} }

#include"susin.cin"

z_type au0(z_type z){ z_type b,d,L; int n,N; DB c[32]=
{ 0.075238095238095238,0.011047619047619048,  0.0025161272589844018,0.00064214089261708309,
 0.00015466661256774635,0.000027370601208859475, -2.7882266241028593e-7,-2.6398530640882673e-6,
 -8.4436657963874166e-7,2.1258409455641387e-7,  2.9625812021061085e-7,5.5084490596214015e-8,
 -8.505934497279864e-8,-5.7880498792647227e-8,  1.931195823640328e-8,4.1375071763898608e-8,
 3.8895251916390044e-9,-2.9644790466835329e-8,  -1.5302998576174588e-8,2.2013832459456463e-8,
 2.5112957261097041e-8,-1.5610882396047076e-8,  -3.8703209933396839e-8,6.3863715618218346e-9,
 6.2295996364195277e-8,1.4512488433891138e-8,  -1.0741668102638471e-7,-7.2006309159592681e-8,
 1.9825395033112708e-7,2.4402848447224951e-7,  -3.8457536962314424e-7,-7.917263057277091e-7};
N=21;  L=log(z); b=z*z; //d=b*(c[N]*.5);
d=b*c[N]*.5;
for(n=N;n>=0;n--) { d+=c[n]; d*=b;}    //d=b*(79./1050.+ b*(.29/2635.));
d+=1.2*L+3./b;
return d-2.08962271972955094;
}

z_type au1(z_type z){ z_type c,d; int m,M;
DB am;
DB a=abs(z); if(a<.7) return au0(z);
m=0;
again: c=sin(z); am=abs(c); if( am<.6 )  return au0(c)-(1.+m); if( am<a ) {a=am; z=c; m++; goto again;}
return au0(c)-1.*m;
}

z_type au9(z_type z){ int m,M;
if(Im(z)<0) return conj(au9(conj(z)));
M=50;  DO(m,M) z=sin(z);
return au0(z)-(0.+M);
}


int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
printf("1.-au1(1)=%19.17f , 1.-au1(z_type(1.,1.e-14))=%19.17f ;1.- susun(1.)=%19.17f, 1.-susun(z_type(1.,1.e-14))=%19.17f \n", 
     1.-Re(au1(1.)),     1.-Re(au1(z_type(1.,1.e-14))),      1.-Re(susin(1.)),     1.-Re(susin(z_type(1.,1.e-14)))     );
//getchar();

int M=548,M1=M+1;
int N=421,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("52.eps","w");  ado(o,332,402);
FILE *o;o=fopen("ausinmapz.eps","w");  ado(o,332,402);
fprintf(o,"1 201 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DB e;
e=2./sinh(1.);
DO(m,M1) { t=(m+.5)/400.; X[m]=e*sinh(1.*t);}
e=2./sinh(1.);
DO(n,N1) { t=(n-210.5)/210.; Y[n]=e*sinh(1.*t);} 
for(m=0;m<4;m+=1){M(m,-2) L(m,2)  }
for(n=-2;n<3;n+=1){M(0,n) L(M_PI,n)}
 fprintf(o,".002 W 0 0 0 RGB S\n");
M(M_PI/2,-2)L(M_PI/2,2)
M(M_PI,-2)L(M_PI,2)
 fprintf(o,".002 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=arcsin(z);
// c=sqrt(3./z);
//   c=susin(z);
   c=au9(z);
//   d=susin(c);
//   d=arcsin(su0(z+1.));
// p=abs(z-d)/(abs(z)+abs(d)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p>-101 && p<101 && q>-101 && q<101){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
 p=200.;q=1.;
/*
  p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.6 ),-p,p); fprintf(o,".001 W .4 1 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(15.  ),-p,p); fprintf(o,".005 W 1 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14.  ),-p,p); fprintf(o,".002 W .2 .2 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(13.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(12.  ),-p,p); fprintf(o,".006 W 0 0 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(11.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(10.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (9.  ),-p,p); fprintf(o,".006 W 0 1 1 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (8.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (7.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (6.  ),-p,p); fprintf(o,".006 W 0 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (5.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (4.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (3.  ),-p,p); fprintf(o,".006 W 1 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (2.  ),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N, (1.  ),-p,p); fprintf(o,".004 W .5 0 0 RGB S\n");
*/
for(m=-4;m<4;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".0016 W 0 .6 0 RGB S\n");
for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".0016 W .9 0 0 RGB S\n");
for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".0016 W 0 0 .9 RGB S\n");
for(m= 1;m<8;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".004 W .8 0 0 RGB S\n");
for(m= 1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".004 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".004 W .5 0 .5 RGB S\n");
for(m=-8;m<9;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".004 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 .01 W S\n");
DO(n,51) {x=-.2*n; M(x-.01,0) L(x-.09,0) }   fprintf(o,"0 0 0 RGB .03 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 ausinmapz.eps"); 
      system(    "open ausinmapz.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 3288pt 
\paperheight 4150pt
\topmargin -100pt
\oddsidemargin -72pt
\textwidth 3200pt
\textheight 4200pt
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\pagestyle{empty}
\begin{document}
\sx{10}{\begin{picture}(328,412)
%\put(10,9){\includegraphics{52}} %extended less details
\put(10,9){\includegraphics{ausinmapz}} %extended
%\put(10,9){\includegraphics{ausin4hmap}} %small
\put(2,406){\sx{1.2}{$y$}}
\put(2,306){\sx{1.2}{$1$}}
\put(2,206){\sx{1.2}{$0$}}
\put(-7,106){\sx{1.2}{$-1$}}
%\put(-7, 06){\sx{1.2}{$-2$}}
\put(9,-1){\sx{1.2}{$0$}}
\put(109,-1){\sx{1.2}{$1$}}
\put(157,-1){\sx{1.2}{$\pi/2$}}
\put(209,-1){\sx{1.2}{$2$}}
\put(309,-1){\sx{1.2}{$3$}}
\put(319,-1){\sx{1.2}{$x$}}
\put(184,208){$v\!=\!0$}
\put(98,200){\rot{90}$u\!=\!2$\ero}
\put(114,200){\rot{90}$u\!=\!1$\ero}
\put(147,186){\rot{41}$u\!=\!0$\ero}
\put(136,131){\rot{-40}$u\!=\!-1$\ero}
\put(125,93){\rot{23}$v\!=\!0$\ero}
\put(92,288){\rot{66}$u\!=\!-2$\ero}
\put(131,282){\rot{31}$u\!=\!-1$\ero}
%\put(159,244){\rot{90}$v\!=\!-0.2$\ero}
%\put(172,250){\rot{90}$v\!=\!0$\ero}
\put(148,260){$u\!=\!-0.5$}
%\put(186,250){\rot{90}$v\!=\!0.2$\ero}
\put(86,158){\rot{90}$v\!=\!3$\ero}
\put(100,155){\rot{90}$v\!=\!2$\ero}
\put(122,148){\rot{90}$v\!=\!1$\ero}
\put(159,144){\rot{90}$v\!=\!0.2$\ero}
\put(172,144){\rot{90}$v\!=\!0$\ero}
\put(186,138){\rot{90}$v\!=\!-0.2$\ero}
\end{picture}}
\end{document}
 

References

Keywords

«Abel function», «Abelfunction», «AuSin», «Special function», «SuSin», «Superfunctions», «Место науки в человеческом знании», «Суперфункции»,