File:SuZexD1mapT.png

Complex map of function SuZex, which is entire superfunction of Zex, $\mathrm{zex}(z)=z\exp(z)$.

$u\!+\!\mathrm i v = \mathrm{SuZex}(c\!+\!\mathrm i y)$

SuZex is built-up at the fixed point zero from asymptotic behavior; the approximation below is implemented:
 * $\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle

f(z)= \frac{-1}{z} +\frac{ \ln(-z)}{2 z^2}+ \frac{-.05\ln(-z)^2-.02\ln(-z)-.4}{z^3}$ For integer $n$,
 * $\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle

F_n(z)=\mathrm{zex}^n\Big( z\!-\!n) \Big)$ The constant $x_n$ is chosen as solution of equation $F_n(x_n)\!=\!1$. Then, the superfunction is evaluated as iteration
 * $\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle

\mathrm{SuZex}(z)=F_n(x_n\!+\!z)$ for integer $n$.

The generator below uses value $n\!=\!16$, which is sufficient to get the camera-ready copy. For the precise computation, more terms in the expansion (1) should be calculated. The precise implementation (with 14 decimal digits) is loaded as SuZex.cin.

C++ generator of curves
using namespace std; typedef complex z_type; // #include "fsexp.cin" //#include "fslog.cin"
 * 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 zex(z_type z){ return z*exp(z);}

int Nitera=16;

z_type suzex0b(z_type z){ return -1./z; } // z_type suzex0(z_type z){ z_type L=log(-z); return (-1.+.5*L/z)/z; } z_type suzex0(z_type z){ z_type L=log(-z); return (-1.+(.5*L + (-.05*L*L-.02*L-.4)/z)/z)/z; }

z_type suzexn(int n, z_type z){int m; z-=0.+n; z=suzex0(z); DO(m,n) z=zex(z); return z; }

main{ int j,k,m,n; DB x1,x,y, p,q, t; z_type z,c,d, cu,cd; x1=-1.04; DO(n,18){ y=Re(suzexn(Nitera,x1)); x=y-1.; x1-=1.5*x; printf("%18.16f %18.16f\n", x1,y);} getchar;

int M=601,M1=M+1; int N=401,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("suZexMap.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-.1); //DO(n,N1) Y[n]=-5.+.02*(n-.5); for(n=0;n.019) { c=suzexn(Nitera,z+x1); p=Re(c); q=Im(c); if(p>-19 && p<19 && fabs(q)>1.e-12 && fabs(p)>1.e-12) 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=2.;q=.3;

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");

//#include "plofu.cin" fprintf(o,"0 setlinejoin 0 setlinecap\n"); fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w); system("epstopdf SuZexMap.eps"); system(   "open SuZexMap.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{b271tMap3}} % %\put(40,20){\ing{ExpMap}} % \put(40,20){\ing{SuZexMap}} % \zoomax % \put(290,611){\sx{4}{$v\!=\!0$}} % \put(183,560){\sx{4}{\rot{90}$u\!=\!0.2$\ero}} % \put(468,560){\sx{4}{\rot{90}$u\!=\!0.4$\ero}} % \put(696,118){\sx{4}{\rot{83}$u\!=\!0$\ero}} % \put(980,236){\sx{4}{\rot{24}$u\!=\!-0.2$\ero}} % \put(790, 44){\sx{4}{\rot{38}$v\!=\!-0.2$\ero}} % \end{picture}} % \end{document} %