File:Autran9tes20t1.jpg

Agreement map for the numerical implementation of function AuTra;

$\mathcal A= A(x+\mathrm i y)$

C++ generator of curves
// Files sutran.cin, ado.cin, conto.cin shold be loaded in order to compile the code below using namespace std; typedef complex z_type; z_type tra(z_type z) {return z+exp(z);}
 * 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"
 * 2) include "arctran.cin"

z_type autran0(z_type z) {z_type e=exp(z); z_type s; int n,M; DB c[20]={-0.166666666666666667, 0.062500000000000, -0.0351851851851851852, 0.0208333333333333333, -0.00976190476190476190, 0.000356867283950617284, 0.00577884857646762409, -0.0054935515873015873, -0.00258505283582444076, 0.0121986400462962963, -0.00649411105018518438, -0.0264514796679871911, 0.0478515524404502325, 0.0537587298747943833, -0.270736261932081259, -0.00655211866410402040, 1.62788126261366988, -1.60768769009409886, -10.8381871746651334, 24.7850929105834429}; M=9; s=c[M]; for(n=M-1;n>=0;n--) { s*=e; s+=c[n];} return z/2.-1./e +e*s + 1.1259817765745026;}

z_type autran(z_type z) { int n; z_type s=z; DB y;   DO(n,40000){ y=fabs(Im(s)); if( y<3. && y < 3.*(-2.5-Re(s)) ) return autran0(s)+(0.+n); s=arctran(s); }  printf("autran: z=%9.6f %9.7f s=%9.6f %9.6f \n",Re(z),Im(z),Re(s),Im(s) ); getchar; return 0.; } int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=251,M1=M+1; int N=1001,N1=N+1; DB X[M1],Y[N1]; DB g[M1*N1]; DB f[M1*N1]; DB w[M1*N1]; char v[M1*N1]; FILE *o;o=fopen("autran9tes20.eps","w"); ado(o,2010,2008); fprintf(o,"1005 1004 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-10.+.08*(m-.5); DO(n,N1) Y[n]=-10.+.02*(n-.5); M(-10.1,3) L(-3.5,3) L(-2.5,0) L(-3.5,-3) L(-10.1,-3) fprintf(o,"C .8 1 .8 RGB F\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]=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=autran(z); c=sutran(c); p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>.5 && p<20.){ g[m*N1+n]=p;} // p=Re(c); q=Im(c); if(p<1000 && p>-1000 && q<1000 && q>-1000 // ( 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=16.;q=.1; conto(o,g,w,v,X,Y,M,N,(16. ),-p,p); fprintf(o,".003 W 1 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(15. ),-p,p); fprintf(o,".012 W .6 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(14. ),-p,p); fprintf(o,".004 W .5 .5 0 RGB S\n"); // conto(o,g,w,v,X,Y,M,N,(13.7 ),-p,p); fprintf(o,".004 W 1 1 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(13. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(12. ),-p,p); fprintf(o,".012 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(11. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(10. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (9. ),-p,p); fprintf(o,".014 W 0 .5 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (8. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (7. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (6. ),-p,p); fprintf(o,".012 W 0 .6 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (5. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (4. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (3. ),-p,p); fprintf(o,".012 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (2. ),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, (1. ),-p,p); fprintf(o,".015 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,".008 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,".008 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,".008 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,".032 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,".032 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".032 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,".03 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n"); M(-10, M_PI)L(-1, M_PI) M(-10,-M_PI)L(-1,-M_PI) fprintf(o,"1 1 1 RGB .044 W S\n"); DO(n,51){M(-1.-.2*n, M_PI)L(-1.-.2*(n+.4), M_PI) } DO(n,51){M(-1.-.2*n,-M_PI)L(-1.-.2*(n+.4),-M_PI) }  fprintf(o,"0 0 0 RGB .05 W S\n");

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf autran9tes20.eps"); system(   "open autran9tes20.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh

c=autran( z_type(-1.,M_PI) ); z=sutran(c); printf("c=%19.16lf %19.16lf z=%19.16lf %19.16lf\n",Re(c),Im(c),Re(z),Im(z)); }

Latex generator of labels
\documentclass[12pt]{article} \paperwidth 2096px \paperheight 2072px \textwidth 2394px \textheight 2300px \topmargin -101px \oddsidemargin -78px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document} \newcommand \zoomax { \put(40,2048){\sx{8}{$y$}} \put(40,1870){\sx{7}{$8$}} \put(40,1670){\sx{7}{$6$}} \put(40,1470){\sx{7}{$4$}} \put(40,1270){\sx{7}{$2$}} \put(40,1070){\sx{7}{$0$}} \put(-12,870){\sx{7}{$-2$}} \put(-12,670){\sx{7}{$-4$}} \put(-12,470){\sx{7}{$-6$}} \put(-12,270){\sx{7}{$-8$}} \put(002, 28){\sx{7}{$-\!10$}} \put(220, 28){\sx{7}{$-8$}} \put(420, 28){\sx{7}{$-6$}} \put(620, 28){\sx{7}{$-4$}} \put(820, 28){\sx{7}{$-2$}} \put(1088, 28){\sx{7}{$0$}} \put(1288, 28){\sx{7}{$2$}} \put(1488, 28){\sx{7}{$4$}} \put(1688, 28){\sx{7}{$6$}} \put(1888, 28){\sx{7}{$8$}} \put(2058, 28){\sx{7}{$x$}} %\put(2166, 28){\sx{7}{$x$}} } \parindent 0pt \sx{1}{\begin{picture}(2102,2094) \zoomax \put(98,88){\ing{autran9tes20}}

\put( 140,1318){\sx{6}{range of}} \put( 140,1224){\sx{6}{the primary}} \put( 140,1130){\sx{6}{approximation}} \put( 910, 940){\sx{9}{\rot{90}$\mathcal A \!\approx \! 16$\ero}} \put(1020, 940){\sx{9}{\rot{90}$\mathcal A \!\approx \! 15$\ero}} %\put( 1440, 940){\sx{9}{\rot{90}$\mathcal A \!\approx \! 14$\ero}} \put( 1700, 940){\sx{9}{\rot{90}$\mathcal A \!\approx \! 14$\ero}} \put(160,610){\sx{8}{$13\!<\! \mathcal A \!<\! 14$}} \put(610,110){\sx{8}{$13\!<\! \mathcal A \!<\! 14$}} \put(1100,410){\sx{9}{$\mathcal A \!\approx \! 14$}}

\end{picture}} \end{document}