File:ArcTaniaMap.png

\[u+\mathrm iv=\mathrm{ArcTania}(x\!+\!\mathrm{i} y)\]

Complex map of the ArcTania function, \[ \mathrm{ArcTania}(z)= z+\ln(z) -1\]

This complex map is used as Fig.5.3 at page 48 of book «Superfunctions» ^[1][2]
in order to invite the reader to practice with complex maps and also to show formalism of superfunctions with the simple example, that actually does not require this formalism to deal with.

C++ generator of curves

//Files ado.cin and conto.cin are necessary to compile the code below

 #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"

 z_type ArcTania(z_type z) {return z + log(z) - 1. ;}

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=160,M1=M+1;
  int N=161,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("arctaniacontour.eps","w");ado(o,162,162);
 fprintf(o,"81 81 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-8.+.1*(m);
 DO(n,80)Y[n]=-8.+.1*n;
        Y[80]=-.033;
        Y[81]= .033;
 for(n=82;n<N1;n++) Y[n]=-8.+.1*(n-1.);
 for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
 for(n=-8;n<9;n++){     M(  -8,n)L(8,n)}
 fprintf(o,".0009 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]; //printf("%5.2f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=ArcTania(z);
 //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
        p=Re(c);q=Im(c);        
        if(p>-99. && p<99. &&
 //       (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
           q>-99. && q<99 //&& fabs(q)> 1.e-19
        )
        {g[m*N1+n]=p;f[m*N1+n]=q;}
                        }}
 fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
 //p=2;q=1;
 for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".011 W 0 .6 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".011 W .9 0 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".011 W 0 0 .9 RGB S\n");
 for(m=1;m<11;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n");
 for(m=1;m<11;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n");
 for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
 y= 0.; for(m=0;m<80;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 1 .5 0 RGB S\n");
 y= 0.; for(m=2;m<80;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf arctaniacontour.eps");
        system(    "open arctaniacontour.pdf");            
        getchar(); system("killall Preview");//for mac
 }

Latex Generator of lables

 % Gerenator of ArcTaniaMap.png %<br>
 % Copyleft 2011 by Dmitrii Kouznetsov %<br>
 \documentclass[12pt]{article} %<br>
 \usepackage{geometry} %<br>
 \usepackage{graphicx} %<br>
 \usepackage{rotating} %<br>
 \paperwidth 854pt %<br>
 \paperheight 844pt %<br>
 \topmargin -96pt %<br>
 \oddsidemargin -98pt %<br>
 \textwidth 1100pt %<br>
 \textheight 1100pt %<br>
 \pagestyle {empty} %<br>
 \newcommand \sx {\scalebox} %<br>
 \newcommand \rot {\begin{rotate}} %<br>
 \newcommand \ero {\end{rotate}} %<br>
 \newcommand \ing {\includegraphics} %<br>
 \begin{document} %<br>
 \sx{5}{ \begin{picture}(164,165) %<br>
 \put(6,5){\ing{arctaniacontour}} %<br>
 \put(2,162){\sx{.7}{$y$}} %<br>
 \put(2,144){\sx{.6}{$6$}} %<br>
 \put(2,124){\sx{.6}{$4$}} %<br>
 \put(2,104){\sx{.6}{$2$}} %<br>
 %\put(23,100){\sx{.8}{$u\!=\!0$}} %<br>
 \put(2, 84){\sx{.6}{$0$}} %<br>
 \put(59,85){\sx{.6}{\bf cut}} %<br>
% \put(20, 84){\sx{.8}{$v\!=\!0$}} %<br>
 \put(-3,64){\sx{.6}{$-2$}} %<br>
 \put(-3,44){\sx{.6}{$-4$}} %<br>
 \put(-3,24){\sx{.6}{$-6$}} %<br>
 \put( 22,0){\sx{.6}{$-6$}} %<br>
 \put( 42,0){\sx{.6}{$-4$}} %<br>
 \put( 62,0){\sx{.6}{$-2$}} %<br>
 \put( 86,0){\sx{.6}{$0$}} %<br>
 \put(106,0){\sx{.6}{$2$}} %<br>
 \put(126,0){\sx{.6}{$4$}} %<br>
 \put(146,0){\sx{.6}{$6$}} %<br>
 \put(164,0){\sx{.7}{$x$}} %<br>

 \put( 81,  23){\rot{81}\sx{.8}{$u\!=\!0$}\ero}%<br>
 \put( 92,  23){\rot{82}\sx{.8}{$u\!=\!1$}\ero}%<br>
 \put(101,  22){\rot{82}\sx{.8}{$u\!=\!2$}\ero}%<br>
 \put(111,  21){\rot{83}\sx{.8}{$u\!=\!3$}\ero}%<br>
 \put(120,  21){\rot{84}\sx{.8}{$u\!=\!4$}\ero}%<br>

 \put(139,155){\rot{4}\sx{.8}{$v\!=\!8$}\ero}%<br>
 \put(138,146){\rot{4}\sx{.8}{$v\!=\!7$}\ero}%<br>
 \put(138,136){\rot{4}\sx{.8}{$v\!=\!6$}\ero}%<br>
 \put(138,127){\rot{4}\sx{.8}{$v\!=\!5$}\ero}%<br>
 \put(137,118){\rot{4}\sx{.8}{$v\!=\!4$}\ero}%<br>
 \put(136,109){\rot{4}\sx{.8}{$v\!=\!3$}\ero}%<br>
 \put(135,100){\rot{4}\sx{.8}{$v\!=\!2$}\ero}%<br>
 \put(134,  92){\rot{3}\sx{.8}{$v\!=\!1$}\ero}%<br>
 \put(134,  84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br>
 \put(134,  76){\rot{-3}\sx{.8}{$v\!=\!-\!1$}\ero}%<br>
 \put(133,  68){\rot{-5}\sx{.8}{$v\!=\!-\!2$}\ero}%<br>
 \put(134,  59){\rot{-5}\sx{.8}{$v\!=\!-\!3$}\ero}%<br>
 \put(135,  51){\rot{-5}\sx{.8}{$v\!=\!-\!4$}\ero}%<br>
 \put(135,  41){\rot{-5}\sx{.8}{$v\!=\!-\!5$}\ero}%<br>
 \put(135,  32){\rot{-5}\sx{.8}{$v\!=\!-\!6$}\ero}%<br>
 \put(136,  23){\rot{-5}\sx{.8}{$v\!=\!-\!7$}\ero}%<br>
 \put(137,  14){\rot{-5}\sx{.8}{$v\!=\!-\!8$}\ero}%<br>
 \end{picture} %<br>
 } %<br>
 \end{document}

References

Keywords

«ArcTania function», «Complex map», «[[]]», «Superfunctions», «Tania function»,