File:Sutralomap.jpg

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) .

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

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 // using namespace std; typedef std::complex z_type; //#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.));}
 * 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"


 * 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.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 //using namespace std; typedef std::complex z_type; //#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.));}
 * 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"


 * 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.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 //using namespace std; typedef std::complex z_type; //#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.));}
 * 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"


 * 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.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}