File:Ausin1susinmapt70.jpg

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Range of validity of representation of function sin through its superfunction SuSin and the Abel function AuSin,

$\sin(z)=\mathrm{SuSin}\big(1+\mathrm{AuSin}(z)\big)$

The range of validity is shown with the complex map in the $z=z+\mathrm i y$ compex plane; within the range, lines $u=\Re(\sin(z))= \mathrm const$ and lines $v=\Im(\sin(z))= \mathrm const$ are drawn.

The range of validity is limited with lines $Im( \mathrm{AuSin}(z))=0$; these lines are also drawn.

C++ generator of the map of sin

// files ado.cin, conto.cin, arcsin.cin, susin.cin, ausin.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 "arcsin.cin"
#include"susin.cin"
#include"ausin.cin"

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

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("ausin1susinmap.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=1.4/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,".001 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=susin(1.+ausin(z));
// 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 && abs(c-sin(z))<.01){ 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,".007 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,".007 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,".007 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,".014 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,".014 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".014 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,".014 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o); free(f); free(g); free(w);
      system("epstopdf ausin1susinmap.eps");
      system( "open ausin1susinmap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

C++ generator of the lines $~ \Im(\mathrm{AuSin}(z))\!=\!0$


#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 "arcsin.cin"
#include"susin.cin"
#include"ausin.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
printf("1.-ausin(1)=%19.17f , 1.-ausin(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(ausin(1.)), 1.-Re(ausin(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("ausinsusinmap2.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,".0004 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,".0003 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=ausin(z);
// c=susin(ausin(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,".012 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 ausinsusinmap2.eps");
      system( "open ausinsusinmap2.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

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