File:ZexD6mapT100.png

Complex map of function zex \(=\) ArcLambertW,

\mathrm{zex}(z)=z\exp(z) \]

\[ u\!+\!\mathrm i v=\mathrm{zex}(x\!+\!\mathrm i y) \]

This map is used as Fig.11.3 at page 136 of book «Superfunctions», 2020 ^[1][2]
in order to show the Transfer function considered in Chapter 11 as an example for the exotic iteration.

C++ generator of curves

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
 using namespace std;
 #include<complex>
 typedef complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
 //#include "fsexp.cin"
 //#include "fslog.cin"

 z_type zex(z_type z){ return z*exp(z);} 

 main(){ int j,k,m,n; DB x1,x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=1201,M1=M+1;
 int N=1201,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("zexD6map.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.+.01*(m-.5);
 DO(n,N1) Y[n]=-6.+.01*(n-.5); 
 //for(n=0;n<N1;n++) { Y[n]=0.6*sinh((3./200.)*(n-200.5)); printf("%3d %9.6f\n",n,Y[n]); }
 for(m=-6;m<7;m++) {M(m,-6)L(m,6)}
 for(n=-6;n<7;n++) {M(  -6,n)L(6,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                        f[m*N1+n]=999;}
 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);  c=zex(z);  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=20.;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");
 fprintf(o,"showpage\n");  fprintf(o,"%c%cTrailer\n",'%','%');
 fclose(o);  free(f); free(g); free(w);
       system("epstopdf zexD6map.eps"); 
       system(    "open zexD6map.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh
 }
<pre>
==[[Latex]] generator of labels==
% <pre>
\documentclass[12pt]{article} % <br>
\paperheight 1228px %  <br>
\paperwidth 1236px %  <br>
\textwidth 1394px %  <br>
\textheight 1300px %  <br>
\topmargin -104px %  <br>
\oddsidemargin -78px %  <br>
\usepackage{graphics} %  <br>
\usepackage{rotating} %  <br>
\newcommand \sx {\scalebox} %  <br>
\newcommand \rot {\begin{rotate}} %  <br>
\newcommand \ero {\end{rotate}} %  <br>
\newcommand \ing {\includegraphics} %  <br>
\newcommand \rmi {\mathrm{i}} %  <br>
\begin{document} %  <br>
\newcommand \zoomax { %  <br>
\put(18,1206){\sx{3.3}{$y$}} %  <br>
\put(18,1113){\sx{3}{$5$}} %  <br>
\put(18,1013){\sx{3}{$4$}} %  <br>
\put(18, 913){\sx{3}{$3$}} %  <br>
\put(18, 813){\sx{3}{$2$}} %  <br>
\put(18, 713){\sx{3}{$1$}} %  <br>
\put(18, 613){\sx{3}{$0$}} %  <br>
\put(-6, 513){\sx{3}{$-1$}} %  <br>
\put(-6, 413){\sx{3}{$-2$}} %  <br>
\put(-6, 313){\sx{3}{$-3$}} %  <br>
\put(-6, 213){\sx{3}{$-4$}} %  <br>
\put(-6, 113){\sx{3}{$-5$}} %  <br>
\put(-6, 013){\sx{3}{$-6$}} %  <br>
\put(014, -5){\sx{3}{$-6$}} %  <br>
\put(114, -5){\sx{3}{$-5$}} %  <br>
\put(214, -5){\sx{3}{$-4$}} %  <br>
\put(314, -5){\sx{3}{$-3$}} %  <br>
\put(414, -5){\sx{3}{$-2$}} %  <br>
\put(514, -5){\sx{3}{$-1$}} %  <br>
\put(635, -5){\sx{3}{$0$}} %  <br>
\put(735, -5){\sx{3}{$1$}} %  <br>
\put(835, -5){\sx{3}{$2$}} %  <br>
\put(935, -5){\sx{3}{$3$}} %  <br>
\put(1035, -5){\sx{3}{$4$}} %  <br>
\put(1135, -5){\sx{3}{$5$}} %  <br>
\put(1227,-4){\sx{3}{$x$}} %  <br>
} %  <br>
\parindent 0pt %  <br>
\sx{1}{\begin{picture}(1252,1220) % <br>
%\put(40,20){\ing{b271tMap3}} %  <br>
%\put(40,20){\ing{ExpMap}} %  <br>
%\put(40,20){\ing{suzexD1map}} %  <br>
\put(40,20){\ing{ZexD6map}} %  <br>
\zoomax %  <br>
\put(90,1162){\sx{4}{\rot{7}$u\!=\!0$\ero}} %  <br>
\put(90,987){\sx{4}{\rot{6}$v\!=\!0$\ero}} %  <br>
\put(90,802){\sx{4}{\rot{5}$u\!=\!0$\ero}} %  <br>
\put(90,611){\sx{4}{$v\!=\!0$}} %  <br>
\put(404,545){\sx{4}{\rot{90}$u\!=\!-0.2$\ero}} %  <br>
\put(980,641){\sx{4}{\rot{-4}$v\!=\!16$\ero}} %  <br>
\put(970,581){\sx{4}{\rot{3}$v\!=\!-16$\ero}} %  <br>
\put(978,517){\sx{4}{\rot{-7}$u\!=\!16$\ero}} %  <br>
\put(970,458){\sx{4}{\rot{-1}$u\!=\!-16$\ero}} %  <br>
% <br>
\put(90,421){\sx{4}{\rot{-5}$u\!=\!0$\ero}} %  <br>
\put(90,237){\sx{4}{\rot{-6}$v\!=\!0$\ero}} %  <br>
\put(90, 61){\sx{4}{\rot{-7}$u\!=\!0$\ero}} %  <br>
\end{picture}} %  <br>
\end{document} %  <br>

References

Keywords

«ArcLambertW», «Exotic iteration», «Exotic iteration», «Fixed point», «LambertW», «Superfunctions», «Transfer function», «Zex»,

«Суперфункции»,