File:Arctra4tesT.jpg

Agreement for the primary approximation app3 of function ArcTra.

$\mathrm{ArcTra}(z)\approx \mathrm{app4}(z)=z-\mathrm e^z$ $ +\mathrm e^{2z}$ $ -\frac{1}{2}\mathrm e^{3z}$

$\displaystyle \mathcal A_4(z)= $ $\displaystyle -\lg \left( \frac{ |\mathrm{tra}(\mathrm{app4}(z))-z| }       { |\mathrm{tra}(\mathrm{app4}(z))| + |z|} \right)$

$\mathcal A=\mathcal A_4(x+\mathrm i y)$ is shown in the $x$, $y$ plane.

C++ generator of curves
// using namespace std; typedef complex z_type; //z_type zex(z_type z){ return z*exp(z);} z_type tra(z_type z){ return z+exp(z);} //#include "Tania.cin" //#include "LambertW.cin" //z_type ArcTra(z_type z){ return z-LambertW(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"

z_type arctra1(z_type z){ static DB c[22]={0.,0.5,-0.0625,.005208333333333333, // 0 - 3 0.0003255208333333333, -0.00021158854166666667, //5 0.00003187391493055556, 1.7680819072420636e-6, //7 -1.8520960732111855e-6, 3.5344398000673434e-7, //9 8.173825669330684e-9, -2.0624513960198102e-8, //11 4.727254469326131e-9, -5.254838295493782e-11, //13 -2.5497322696797093e-10,6.897078914225712e-11, //15 -3.004636375311634e-12,-3.313983848679892e-12, //17 1.054773386200646e-12,-7.811289918947213e-14, //19 -4.391790189701172e-14, 1.6573692169992586e-14}; //21 z_type t=z-1.; z_type s=c[21]*z; int n; for(n=20;n>0;n--){ s+=c[n]; s*=t;} return s;}

z_type arctra1i3(z_type z) { z_type t=arctra1(z); t+=(z-t-exp(t))/(1.+exp(t)); t+=(z-t-exp(t))/(1.+exp(t)); t+=(z-t-exp(t))/(1.+exp(t)); return t; }

z_type arctra2(z_type z) { z_type s,t; t=z+z_type(1.,-M_PI); t/=2.; t=sqrt(t); //s= I*M_PI - z_type(0.,-2.)*t; //s= I*M_PI + t*(z_type(0.,2.)+t*(2./3)); s= I*M_PI +t*( z_type(0.,-2.) +t*( 2./3 +t*( z_type(0.,2./9.) +t*( -8./135. +t*( z_type(0.,-1./135.) +t*( -32./8505. +t*( z_type(0.,-139./42525.) +t*( 32./25515. +t*( z_type(0.,571./4592700.) +t*( 35968./189448875. )))))))))); return s; }

z_type arctra3(z_type z) { z_type c[5]; z_type L=log(z); z_type s; int n,M=4; c[0]= -1.; c[1]= 1. + L*(-1./2. ); c[2]= -1. + L*( 1.5 + L*( -1./3.)) ; c[3]= 1. + L*(-3. + L*( 11./6. + L*(-1./4.))) ; c[4]= -1. + L*( 5. + L*(-35./6. + L*(25./12. + L*(-1./5.)))); s=c[M]; for(n=M-1;n>=0;n--){ s/=z; s+=c[n];} //s=-1.; // s=c[0]+c[1]/z; return L*(1.+s/z); }

z_type arctra4(z_type z) { z_type e=exp(z); return z +e*(-1. +e*( 1. +e*(-0.5) )); }

int main{ int j,k,m,n; DB x1,x,y, p,q, t; z_type z,c,d, cu,cd; int M=601,M1=M+1; int N=601,N1=N+1; //int N=801,N1=N+1; DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array. char *v; 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))); v=(char *)malloc((size_t)((M1*N1)*sizeof(char))); //char v[M1*N1]; // v is working array FILE *o;o=fopen("arctra4tes.eps","w"); ado(o,1202,1202); fprintf(o,"601 601 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-6.+.02*(m-.5); DO(n,N1) Y[n]=-6.+.02*(n-.5); //DO(n,N1) Y[n]=-6.+.0151*(n-.5); //for(n=0;n.1 && p<19. ) g[m*N1+n]=p; //       if(p>-19 && p<19 &&  fabs(q)>1.e-12 && fabs(p)>1.e-12) //f[m*N1+n]=q; }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.;q=.1; /* 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. ),-p,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");

M(-6, M_PI)L(-1, M_PI) M(-6,-M_PI)L(-1,-M_PI) fprintf(o,".01 W 1 1 1 RGB S\n");

conto(o,g,w,v,X,Y,M,N,(15. ),-2,2); fprintf(o,".03 W .7 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(14. ),-2,2); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(13. ),-2,2); fprintf(o,".01 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(12. ),-1,1); fprintf(o,".04 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,".04 W 0 .8 0 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 1 0 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,".04 W 0 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,".06 W 1 .5 0 RGB S\n");

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w); system("epstopdf arctra4tes.eps"); system(   "open arctra4tes.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh } //

Latex generator of labels
% \documentclass[12pt]{article} \paperheight 1228px \paperwidth 1236px \textwidth 1394px \textheight 1300px \topmargin -104px \oddsidemargin -78px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document} \newcommand \zoomax { \put(18,1206){\sx{3.3}{$y$}} \put(18,1113){\sx{3}{$5$}} \put(18,1013){\sx{3}{$4$}} \put(18, 913){\sx{3}{$3$}} \put(18, 813){\sx{3}{$2$}} \put(18, 713){\sx{3}{$1$}} \put(18, 613){\sx{3}{$0$}} \put(-6, 513){\sx{3}{$-1$}} \put(-6, 413){\sx{3}{$-2$}} \put(-6, 313){\sx{3}{$-3$}} \put(-6, 213){\sx{3}{$-4$}} \put(-6, 113){\sx{3}{$-5$}} \put(-6, 013){\sx{3}{$-6$}} \put(014, -5){\sx{3}{$-6$}} \put(114, -5){\sx{3}{$-5$}} \put(214, -5){\sx{3}{$-4$}} \put(314, -5){\sx{3}{$-3$}} \put(414, -5){\sx{3}{$-2$}} \put(514, -5){\sx{3}{$-1$}} \put(635, -5){\sx{3}{$0$}} \put(735, -5){\sx{3}{$1$}} \put(835, -5){\sx{3}{$2$}} \put(935, -5){\sx{3}{$3$}} \put(1035, -5){\sx{3}{$4$}} \put(1135, -5){\sx{3}{$5$}} \put(1227,-4){\sx{3}{$x$}} } \parindent 0pt \sx{1}{\begin{picture}(1252,1220) %\put(40,20){\ing{ArcTraMap}} \zoomax %\put(40,20){\ing{arctra1tes2}} %\put(40,20){\ing{arctra2tes}} %\put(40,20){\ing{arctra3tes}} \put(40,20){\ing{arctra4tes}}

\put(276,444){\sx{8}{\rot{90}$\mathcal A\!=\!6$\ero}} \put(490,444){\sx{8}{\rot{90}$\mathcal A\!=\!3$\ero}} \put(630,444){\sx{8}{\rot{90}$\mathcal A\!=\!1$\ero}} \put(860,540){\sx{8}{$\mathcal A\!<\!1$}} \end{picture}} \end{document}