File:Sutralomap.jpg

From TORI
Jump to: navigation, search
Original file ‎(2,533 × 1,254 pixels, file size: 1.26 MB, MIME type: image/jpeg)

Overlappings of complex map of function

$z \mapsto -\ln(-z)$

with two its approximations with entire functions,

$f_2(z)=\mathrm{SuTra}(2z) + \ln(2)$ , left, and

$f_4(z)=\mathrm{SuTra}(4z) + \ln(4)$ , right.

for function $f$, levels are shown with

$u\!+\!\mathrm i v= f(x\!+\!\mathrm i y)$

in the $x$, $y$ plane.

This image is used as figure 20.7 of the Book Superfunction (In Russian, 2014; the English version is in preparation, 2015) [1].

Function SuTra as entire function with logarithmic asymptotic is described also in year 2013 at Hikari [2].

References

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014
  2. ↑ http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf D.Kouznetsov. Entire Function with Logarithmic Asymptotic. Applied Mathematical Sciences, Vol. 7, 2013, no. 131, 6527 - 6541


C++ generator of map of $z\mapsto -\ln(-z)$

Files ado.cin, conto.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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
//#include "tania.cin"
//#include "LambertW.cin"
//#include "SuZex.cin"
//#include "AuZex.cin"
//z_type tra(z_type z) {return z+exp(z);}
// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
//z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
// return tra(sutra(z-1.));}

#include"sutran.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
int M=1001,M1=M+1;
int N=1001,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("sutranmap.eps","w"); ado(o,2002,2002);
FILE *o;o=fopen("mlogmap.eps","w"); ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.02*(m-.5);
DO(n,N1) Y[n]=-10+.02*(n-.5);
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-10;m<11;m++){M(m,-10) L(m,10) }
for(n=-10;n<11;n++){M( -10,n) L(10,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){ g[m*N1+n]=999;
                       f[m*N1+n]=999;}
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=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=log(suzex(z));
// c=sutran(z);
  c=-log(-z);
// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/* p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 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,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,".01 W .2 .2 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,".03 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,".03 W 0 1 1 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,".04 W 0 .5 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,".02 W 1 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 .5 0 0 RGB S\n");
*/
for(m=-8;m<8;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<8;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<8;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<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 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 .02 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 mlogmap.eps");
      system( "open mlogmap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

C++ generator of map of $f_2$

Files ado.cin, conto.cin, sutran.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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
//#include "tania.cin"
//#include "LambertW.cin"
//#include "SuZex.cin"
//#include "AuZex.cin"
//z_type tra(z_type z) {return z+exp(z);}
// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
//z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
// return tra(sutra(z-1.));}

#include"sutran.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
int M=1001,M1=M+1;
int N=1001,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("sutra2map.eps","w"); ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.02*(m-.5);
DO(n,N1) Y[n]=-10+.02*(n-.5);
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-10;m<11;m++){M(m,-10) L(m,10) }
for(n=-10;n<11;n++){M( -10,n) L(10,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){ g[m*N1+n]=999;
                       f[m*N1+n]=999;}
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=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=log(suzex(z));
  c=sutran(2.*z)+log(2.);
// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/* p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 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,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,".01 W .2 .2 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,".03 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,".03 W 0 1 1 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,".04 W 0 .5 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,".02 W 1 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 .5 0 0 RGB S\n");
*/
for(m=-8;m<8;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<8;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<8;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<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 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 .02 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 sutra2map.eps");
      system( "open sutra2map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

C++ generator of map of $f_4$

Files ado.cin, conto.cin, sutran.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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
//#include "tania.cin"
//#include "LambertW.cin"
//#include "SuZex.cin"
//#include "AuZex.cin"
//z_type tra(z_type z) {return z+exp(z);}
// z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
//z_type sutra(z_type z){ if( Re(z)<2. ) return log(suzex(z));
// return tra(sutra(z-1.));}

#include"sutran.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
//DB x1=-1.1259817765745026; DO(n,8){ y=Re(suzex(x1)); x=y-1.; x1+=-1.2*x; printf("%18.16f %18.16f\n", x1,y);} getchar();
int M=1001,M1=M+1;
int N=1001,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("sutra4map.eps","w"); ado(o,2002,2002);
fprintf(o,"1001 1001 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-10+.02*(m-.5);
DO(n,N1) Y[n]=-10+.02*(n-.5);
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-10;m<11;m++){M(m,-10) L(m,10) }
for(n=-10;n<11;n++){M( -10,n) L(10,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){ g[m*N1+n]=999;
                       f[m*N1+n]=999;}
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=AuZex01(z-1.);
// c=AuZexAsy(LambertW(z))+1.;
//c=log(suzex(z));
  c=sutran(4.*z)+log(4.);
// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p>-19 && p<19 && ( x<2. || (fabs(q)>1.e-12 && fabs(p)>1.e-12)) ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;
/* p=9;q=.16;
 conto(o,g,w,v,X,Y,M,N,(15.3 ),-p,p); fprintf(o,".01 W .4 1 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,(14.7 ),-p,p); fprintf(o,".01 W 1 .5 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,".01 W .2 .2 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,".03 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,".03 W 0 1 1 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,".04 W 0 .5 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,".02 W 1 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 .5 0 0 RGB S\n");
*/
for(m=-8;m<8;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<8;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<8;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<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 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 .02 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 sutra4map.eps");
      system( "open sutra4map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex combiner


\documentclass[12pt]{article}
\paperwidth 1216px
\paperheight 602px
\textwidth 1380px
\textheight 1300px
\topmargin -106px
\oddsidemargin -71px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
\parindent 0pt
\sx{.3}{\begin{picture}(2000,2002)
\put(30,1950){\sx{8}{$y$}}
\put(30,1780){\sx{7}{$8$}}
\put(30,1580){\sx{7}{$6$}}
\put(30,1380){\sx{7}{$4$}}
\put(30,1180){\sx{7}{$2$}}
\put(30,0980){\sx{7}{$0$}}
\put(-22,780){\sx{7}{$-2$}}
\put(-22,580){\sx{7}{$-4$}}
\put(-22,380){\sx{7}{$-6$}}
\put(-22,180){\sx{7}{$-8$}}
\put(120, 6){\sx{7}{$-8$}}
\put(320, 6){\sx{7}{$-6$}}
\put(520, 6){\sx{7}{$-4$}}
\put(720, 6){\sx{7}{$-2$}}
\put(988, 6){\sx{7}{$0$}}
\put(1188, 6){\sx{7}{$2$}}
\put(1388, 6){\sx{7}{$4$}}
\put(1588, 6){\sx{7}{$6$}}
\put(1788, 6){\sx{7}{$8$}}
\put(1958, 6){\sx{8}{$x$}}
%\put(0,0){\ing{SuTraMap}}
\put(0,0){\ing{sutra2map}}
\put(0,0){\ing{mlogmap}}
%\zoomax %
\put(200,1414){\sx{7}{\rot{56}$u\!=\!-2.2$\ero}}
\put(352,1328){\sx{7}{\rot{56}$u\!=\!-2$\ero}}
\put(462,1238){\sx{7}{\rot{56}$u\!=\!-1.8$\ero}}
\put(1277,1590){\sx{7}{\rot{68}$v\!=\!2$\ero}}
\put(596,1632){\sx{7}{\rot{-55}$v\!=\!1$\ero}}
\put(340,1128){\sx{7}{\rot{-10}$v\!=\!0.2$\ero}}
\put(310,0984){\sx{7}{$v\!=\!0$}}
\put(330,0832){\sx{7}{\rot{10}$v\!=\!-0.2$\ero}}
\put(620,0332){\sx{7}{\rot{55}$v\!=\!-1$\ero}}
\put(1209,488){\sx{7}{\rot{-70}$v\!=\!-2$\ero}}
%\put(1490,772){\sx{7}{\rot{-11}$v\!=\!-3$\ero}}
\end{picture}}
\rule{8pt}{0pt}
%\sx{.12}{\begin{picture}(2002,2002)
\sx{.3}{\begin{picture}(2000,2002)
%\put(40,20){\ing{b271tMap3}}
%\put(40,20){\ing{ExpMap}}
\put(30,1950){\sx{8}{$y$}}
\put(30,1780){\sx{7}{$8$}}
\put(30,1580){\sx{7}{$6$}}
\put(30,1380){\sx{7}{$4$}}
\put(30,1180){\sx{7}{$2$}}
\put(30,0980){\sx{7}{$0$}}
\put(-22,780){\sx{7}{$-2$}}
\put(-22,580){\sx{7}{$-4$}}
\put(-22,380){\sx{7}{$-6$}}
\put(-22,180){\sx{7}{$-8$}}
\put(120, 6){\sx{7}{$-8$}}
\put(320, 6){\sx{7}{$-6$}}
\put(520, 6){\sx{7}{$-4$}}
\put(720, 6){\sx{7}{$-2$}}
\put(988, 6){\sx{7}{$0$}}
\put(1188, 6){\sx{7}{$2$}}
\put(1388, 6){\sx{7}{$4$}}
\put(1588, 6){\sx{7}{$6$}}
\put(1788, 6){\sx{7}{$8$}}
\put(1958, 6){\sx{8}{$x$}}
%\put(0,0){\ing{SuTraMap}}
\put(0,0){\ing{sutra4map}}
\put(0,0){\ing{mlogmap}}
%\zoomax %
\put(200,1414){\sx{7}{\rot{56}$u\!=\!-2.2$\ero}}
\put(352,1328){\sx{7}{\rot{56}$u\!=\!-2$\ero}}
\put(462,1238){\sx{7}{\rot{56}$u\!=\!-1.8$\ero}}
\put(1277,1590){\sx{7}{\rot{68}$v\!=\!2$\ero}}
\put(596,1632){\sx{7}{\rot{-55}$v\!=\!1$\ero}}
\put(340,1128){\sx{7}{\rot{-10}$v\!=\!0.2$\ero}}
\put(310,0984){\sx{7}{$v\!=\!0$}}
\put(330,0832){\sx{7}{\rot{10}$v\!=\!-0.2$\ero}}
\put(620,0332){\sx{7}{\rot{55}$v\!=\!-1$\ero}}
\put(1209,488){\sx{7}{\rot{-70}$v\!=\!-2$\ero}}
%\put(1490,772){\sx{7}{\rot{-11}$v\!=\!-3$\ero}}
\end{picture}}
\end{document}

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current06:14, 1 December 2018Thumbnail for version as of 06:14, 1 December 20182,533 × 1,254 (1.26 MB)Maintenance script (talk | contribs)Importing image file
  • You cannot overwrite this file.

There are no pages that link to this file.

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.
Retrieved from "https://mizugadro.mydns.jp/t/index.php?title=File:Sutralomap.jpg&oldid=21755"