File:Autran0m9tes64t.jpg

Agreement $\mathcal A=A(x+\mathrm i y)$ for the primary approximation $g$ of function AuTra

with $M=9$.

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"

//#include "tania.cin" //#include "LambertW.cin" //#include "auzex.cin"

z_type arctra(z_type z){return z-Tania(z-1.);} z_type autra0(z_type z) {return auzex(exp(z));} z_type autra(z_type z) { if(fabs(Im(z))<M_PI) return auzex(exp(z)); return autra(arctra(z)) +1.; }


 * 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=14; 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,400){ y=fabs(Im(s)); if( y<3. && y < 1.5*(-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=200,M1=M+1; int N=1600,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("autran0m9tes64.eps","w"); ado(o,602,802); fprintf(o,"501 401 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-5.+.028*(m-.5); DO(n,N1) Y[n]=-4.+.005*(n-.5);

M(-5.1,3) L(-3.5,3) L(-2.5,0) L(-3.5,-3) L(-5.1,-3) fprintf(o,"C .7 1 .7 RGB F\n");

for(m=-5;m<2;m++){M(m,-4) L(m,4) } for(n=-4;n<5;n++){M( -5,n) L(1,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=autran0(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 0 0 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"); //#include "plofu.cin"

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); // free(f); free(g); free(w); system("epstopdf autran0m9tes64.eps"); system(   "open autran0m9tes64.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh

c=autra( 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 602px \paperheight 802px \textwidth 3004px \textheight 3000px \topmargin -106px \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} \parindent 0pt \hskip 6pt \begin{picture}(610,802) %\put(1,1){\ing{logq2map}} \put(0,0){\ing{autran0m9tes64}} \put(18,780){\sx{4}{$y$}} \put(18,688){\sx{4}{$3$}} \put(18,588){\sx{4}{$2$}} \put(18,488){\sx{4}{$1$}} \put(18,388){\sx{4}{$0$}} \put(-6,288){\sx{4}{$-1$}} \put(-6,188){\sx{4}{$-2$}} \put(-6,088){\sx{4}{$-3$}} %\put(590,2){\sx{.7}{\rot{80}$v\!=\!4$\ero}} \put(078,3){\sx{3}{$-4$}} \put(178,3){\sx{3}{$-3$}} \put(278,3){\sx{3}{$-2$}} \put(378,3){\sx{3}{$-1$}} \put(494,3){\sx{3}{$0$}} \put(580,3){\sx{3.6}{$x$}}

\put(210,750){\sx{2.8}{$\mathcal A \!<\! 1$}} \put(276,350){\sx{3}{\rot{90}$\mathcal A\!=\!15$\ero}} \put(332,350){\sx{3}{\rot{90}$\mathcal A\!=\!12$\ero}} \put(386,360){\sx{3}{\rot{90}$\mathcal A\!=\!9$\ero}} \put(440,360){\sx{3}{\rot{90}$\mathcal A\!=\!6$\ero}} \put(500,260){\sx{3}{\rot{90}$\mathcal A\!=\!3$\ero}} \put(542,260){\sx{3}{\rot{90}$\mathcal A\!=\!1$\ero}} \put(528,390){\sx{2.8}{$\mathcal A \!<\! 1$}}

%\put(111,412){\sx{3}{$\mathcal A > 16$}} \put(166,142){\sx{2.8}{$\mathcal A \!\approx\! 16$}} %\put(530,155){\sx{2.8}{$\mathcal A \!<\! 1$}} \end{picture} \end{document}