File:Vladi02.jpg

Complex map of arctetration ate;

$u\!+\!\mathrm i v=\mathrm{ate}(x\!+\!\mathrm i y)$

Usage: this is figure 15.1 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

First time published in the Vladikavkaz Matehmatical Journal .

C++ generator of map
ado.cin, conto.cin, fslog.cin should be loaded in order to compile the code below.

typedef std::complex z_type; //#include "superlo.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; //z_type Zo=z_type(.31813150520476413, 1.3372357014306895); //z_type Zc=z_type(.31813150520476413,-1.3372357014306895);
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 1) include "fslog.cin"
 * 2) include "conto.cin"

int M=400,M1=M+1; int N=364,N1=N+1; //DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. DB X[401],Y[365], g[146365],f[146365], w[146365]; // w is working array. //DB X[401],Y[365], g[160000],f[16000], w[16000]; // w is working array. //char v[M1*N1]; // v is working array char v[146365]; // v is working array FILE *o;o=fopen("vladi02c.eps","w");ado(o,202,122); fprintf(o,"101 61 translate\n 10 10 scale\n");

z_type L=z_type(.31813150520476413, 1.3372357014306895); p=Re(L); q=Im(L); DB R=abs(L); DB A=arg(L); fprintf(o,"0 0 %9.6f %9.6f %9.6f arc C .6 1 .5 RGB F\n",R,-180/M_PI*A,180/M_PI*A);

DB sx=8./sinh(.01*M); DO(m,M1) X[m]=sx*sinh(.02*(m-M/2)); for(n=0;n-Im(L)) break; Y[n]=y;} m=n; y=-Im(L)-.001; Y[m]=y; m++; y=-Im(L)+.001; Y[m]=y; for(n=m+1;nIm(L)) break; Y[n]=y;} m=n; y=Im(L)-.001; Y[m]=y; m++; y=Im(L)+.001; Y[m]=y; for(n=m+1;n-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8)      g[m*N1+n]=p; if(q>-999 && q<999 && fabs(q)> 1.e-8)                          f[m*N1+n]=q; }}

p=.4;q=.4; //#include"plofu.cin" for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".01 W 0 .6 0 RGB S\n"); for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q); fprintf(o,".01 W .9 0 0 RGB S\n"); for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q); fprintf(o,".01 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n"); for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .5 0 .5 RGB S\n"); for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");

M(Re(L), Im(L)) L(-10, Im(L)) M(Re(L),-Im(L)) L(-10,-Im(L)) fprintf(o,".02 W 1 1 1 RGB S\n");

DO(m,17){ M(Re(L)-.5*m, Im(L)) L(Re(L)-.5*(m+.5), Im(L))} DO(m,17){ M(Re(L)-.5*m,-Im(L)) L(Re(L)-.5*(m+.5),-Im(L))} fprintf(o,".08 W 0 0 0 RGB S\n");

// fprintf(o,".1 W 0 0 0 RGB [.12 .14] 1 setdash S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf vladi02c.eps"); system(   "open vladi02c.pdf"); //getchar; system("killall Preview");//macintosh }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{rotating} \usepackage{geometry} \paperwidth 418px %\paperheight 134px \paperheight 292px \topmargin -104pt \oddsidemargin -94pt \pagestyle{empty} \begin{document} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \parindent 0pt \hskip -14pt \sx{2.5}{\begin{picture}(220,120) %\put(0,0){\includegraphics{figslogG}} \put(0,0){\includegraphics{vladi02c}} \put(17,114){\sx{.6}{$y$}} \put(17, 99){\sx{.58}{$4$}} \put(17, 89){\sx{.58}{$3$}} \put(17, 79){\sx{.58}{$2$}} \put(17, 69){\sx{.58}{$1$}} \put(17, 59){\sx{.58}{$0$}} \put(12, 49){\sx{.58}{$-1$}} \put(12, 39){\sx{.58}{$-2$}} \put(12, 29){\sx{.58}{$-3$}} \put(12, 19){\sx{.58}{$-4$}} \put(12, 9){\sx{.58}{$-5$}} \put(178,5){\sx{.6}{$x$}} %\put(180,4){\sx{.6}{$8$}} \put(160,5.2){\sx{.6}{$6$}} \put(140,5.2){\sx{.6}{$4$}} \put(120,5.2){\sx{.6}{$2$}} \put( 99.8,5.2){\sx{.6}{$0$}} \put( 76,5.2){\sx{.6}{$-2$}} \put( 56,5.2){\sx{.6}{$-4$}} \put( 36,5.2){\sx{.6}{$-6$}} %\put(105,57){\sx{1}{$G$}}

\put( 64,118){\sx{.5}{$u\!=\!2.2$}} \put( 94,118){\sx{.5}{$u\!=\!2$}} \put(118,118){\sx{.5}{$v\!=\!0.6$}} \put(148,118){\sx{.5}{$v\!=\!0.4$}} %\put(182,100){\sx{.5}{$v\!=\!0.2$}} %\put(182, 92){\sx{.5}{$u\!=\!1.8$}} \put(166, 59.8){\sx{.5}{$v\!=\!0$}} \end{picture}} \end{document}