File:AuTraMapT.jpg

Complex map of function AuTra, $u\!+\!i v\!=\! \mathrm{AuTra}(x\!+\!\mathrm i y)$

AuTra is Abel function of the Trappmann function $\mathrm{tra}(z)\!=\!z\!+\!\exp(z)$.

C++ generator of curves
// Files ado.cin, conto.cin, tania.cin, LambertW.cin, SuZex.cin, AuZex.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 "tania.cin"
 * 6) include "LambertW.cin"
 * 7) include "SuZex.cin"
 * 8) include "AuZex.cin"

z_type arctra(z_type z){return z-Tania(z-1.);} z_type tra(z_type z) {return z+exp(z);} 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.; }

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=501,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("AuTraMap.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+.04*(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=auzex(z); //c=suzex(c); c=autra(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<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=2.; /* 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,".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 AuTraMap.eps"); system(   "open AuTraMap.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of labels
%File AuTraMap.pdf should be generated with the code above in order to compile the Latex document below.

% % \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) % %\put(40,20){\ing{b271tMap3}} % %\put(40,20){\ing{ExpMap}} % \put(98,88){\ing{AuTraMap}} % \zoomax % \put(218,1384){\sx{9}{\bf cut}} % \put(218, 752){\sx{9}{\bf cut}} % % \put( 224,1836){\sx{8}{\rot{35}$v\!=\!0.6$\ero}} \put( 1590,1496){\sx{8}{\rot{70}$v\!=\!0.4$\ero}} \put( 1800,1360){\sx{8}{\rot{38}$v\!=\!0.2$\ero}} \put( 140,1068){\sx{9}{$v\!=\!0$}} % \put(1874,1068){\sx{9}{$v\!=\!0$}} % \put( 1740, 814){\sx{8}{\rot{-40}$v\!=\!-0.2$\ero}} \put( 194, 306){\sx{8}{\rot{-35}$v\!=\!-0.6$\ero}} % \put( 606, 1920){\sx{9}{\rot{-64}$u\!=\!3$\ero}} \put( 1500, 992){\sx{8.2}{\rot{90}$u\!=\!2$\ero}} \put( 1630, 960){\sx{8}{\rot{90}$u\!=\!2.2$\ero}} \put( 1846, 960){\sx{8}{\rot{90}$u\!=\!2.4$\ero}} \put( 1724, 400){\sx{8}{\rot{22}$u\!=\!2.6$\ero}} \put( 670, 290){\sx{9}{\rot{60}$u\!=\!3$\ero}} % \end{picture}} % \end{document} %