File:Ausin1susinmapt70.jpg

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.

using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "arcsin.cin"
 * 6) include"susin.cin"
 * 7) 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 $~ \Im(\mathrm{AuSin}(z))\!=\!0$
using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "arcsin.cin"
 * 6) include"susin.cin"
 * 7) 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 }