File:SdPow2map.jpg

Complex map of function SdPow$_2$ ;

$F(z)=$SdPow$_2(z)=\exp(-2^z)$

is shown with lines of constant real part and lines of constant imaginary part:

$u\!+\mathrm i v=F(x\!+\mathrm i y)$

Superpower
Superpower function $F$ is superfunction for the quadratic transfer function $T(z)\!=\!z^2$, which is in its turn, interpreted as power function to base 2.

Superpower function $F$ is solution of the transfer equation

$F(z\!+\!1)=T(F(z))$

C++ generator of map
Files ado.cin and conto.cin should be loaded in order to compile the C++ code below. typedef std::complex z_type; int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=501,M1=M+1; int N=501,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("superpower2ma.eps","w");ado(o,1020,1020); FILE *o;o=fopen("sdpow2ma.eps","w");ado(o,1020,1020); fprintf(o,"510 510 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-5.+.02*(m-.5); DO(n,N1) Y[n]=-5.+.02*(n-.5); for(m=-5;m<6;m++){if(m==0){M(m,-5.1)L(m,5.1)} else{M(m,-5)L(m,5)}} for(n=-5;n<6;n++){ M( -5,n)L(5,n)} fprintf(o,".01 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=1.e15; f[m*N1+n]=1.e15;} DO(m,M1){x=X[m]; //printf("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=exp(-pow(2.,z)); p=Re(c); q=Im(c); if(p>-2.e15 && p<1.e15      && q>-1.e15 && q<1.e15       ) {g[m*N1+n]=p; f[m*N1+n]=q; }                     }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=10;q=.4; for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".013 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,".028 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,".028 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-1.e14,1.e14); fprintf(o,".028 W .6 0 .6 RGB S\n"); for(m=-10;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n"); m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-1.e14,1.e14); fprintf(o,".024 W 0 0 0 RGB S\n"); for(m=1;m<11;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");
 * 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.)
 * 4) include "conto.cin"

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sdpow2ma.eps"); system( "open sdpow2ma.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{geometry} %\usepackage{rotate} %\usepackage{rotation} \usepackage{rotating} \paperwidth 1044px \paperheight 1036px \topmargin -98px \oddsidemargin -90px \textwidth 2000px \textheight 2000px \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(1030,1026) \normalsize \put(30,20){\ing{"sdpow2ma"}} %\put(30,20){\ing{"superpower2ma"}} \put(16,1016){\sx{3}{$y$}} \put(16,918){\sx{3}{$4$}} \put(16,818){\sx{3}{$3$}} \put(16,718){\sx{3}{$2$}} \put(16,618){\sx{3}{$1$}} \put(16,518){\sx{3}{$0$}} \put(-10,418){\sx{3}{$-1$}} \put(-10,318){\sx{3}{$-2$}} \put(-10,218){\sx{3}{$-3$}} \put(-10,118){\sx{3}{$-4$}} \put(-10,18){\sx{3}{$-5$}} \put(10,-1){\sx{3}{$-5$}} \put(110,-1){\sx{3}{$-4$}} \put(210,-1){\sx{3}{$-3$}} \put(310,-1){\sx{3}{$-2$}} \put(410,-1){\sx{3}{$-1$}} \put(534,-1){\sx{3}{$0$}} \put(634,-1){\sx{3}{$1$}} \put(734,-1){\sx{3}{$2$}} \put(834,-1){\sx{3}{$3$}} \put(934,-1){\sx{3}{$4$}} \put(1028,-1){\sx{3.1}{$x$}} \put(150,978){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero} \put(150,756){\rot{1}{ \sx{3}{$u\!=\!1$}} \ero} \put(150,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero} \put(150,290){\rot{-1}{ \sx{3}{$u\!=\!1$}} \ero} \put(150,68){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero} % \put(336,484){\rot{90}{ \sx{3}{$u\!=\!0.8$}} \ero} \put(454,484){\rot{90}{ \sx{3}{$u\!=\!0.6$}} \ero} \put(538,484){\rot{90}{ \sx{3}{$u\!=\!0.4$}} \ero} \put(618,484){\rot{90}{ \sx{3}{$u\!=\!0.2$}} \ero} %\put(530,489){\rot{90}{ \sx{3}{$u\!=\!2$}} \ero} % \put(780,564){\rot{-11}{ \sx{3}{$u\!=\!0$}} \ero} \put(790,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero} \put(782,482){\rot{10}{ \sx{3}{$u\!=\!0$}} \ero} % \put(310,926){\rot{90}{ \sx{3}{$u\!=\!1.2$}} \ero} \put(400,926){\rot{90}{ \sx{3}{$u\!=\!1.4$}} \ero} \put(442,926){\rot{90}{ \sx{3}{$u\!=\!1.6$}} \ero} \put(476,926){\rot{90}{ \sx{3}{$u\!=\!1.8$}} \ero} \put(502,936){\rot{90}{ \sx{3}{$u\!=\!2$}} \ero} % \put(800,978){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}%%% \put(788,938){\rot{12}{ \sx{3}{$u\!=\!0$}} \ero} % \put(816,906){\rot{24}{ \sx{3}{$v\!=\!0$}} \ero} % \put(820,870){\rot{32}{ \sx{3}{$u\!=\!0$}} \ero} % \put(310,26){\rot{90}{ \sx{3}{$u\!=\!1.2$}} \ero} \put(400,26){\rot{90}{ \sx{3}{$u\!=\!1.4$}} \ero} \put(442,26){\rot{90}{ \sx{3}{$u\!=\!1.6$}} \ero} \put(476,26){\rot{90}{ \sx{3}{$u\!=\!1.8$}} \ero} \put(502,36){\rot{90}{ \sx{3}{$u\!=\!2$}} \ero} % \put(658,172){\rot{-31}{ \sx{3}{$u\!=\!0$}} \ero} % % \put(508,800){\rot{-33}{ \sx{3}{$v\!=\!-1$}} \ero} %\put(480,748){\rot{-16}{ \sx{3}{$v\!=\!-0.8$}} \ero} \put(476,730){\rot{-17}{ \sx{3}{$v\!=\!-0.6$}} \ero} \put(478,670){\rot{-6}{ \sx{3}{$v\!=\!-0.4$}} \ero} \put(480,602){\rot{0}{ \sx{3}{$v\!=\!-0.2$}} \ero}

\put(480,446){\rot{2}{ \sx{3}{$v\!=\!0.2$}} \ero}%% \put(482,372){\rot{9}{ \sx{3}{$v\!=\!0.4$}} \ero}%% \put(500,320){\rot{16}{ \sx{3}{$v\!=\!0.6$}} \ero} \put(528,262){\rot{28}{ \sx{3}{$v\!=\!1$}} \ero} %\put(508,348){\rot{-33}{ \sx{3}{$v\!=\!-1$}} \ero} %\put(480,268){\rot{-12}{ \sx{3}{$v\!=\!-0.6$}} \ero} %\put(480,210){\rot{-4}{ \sx{3}{$v\!=\!-0.4$}} \ero} %\put(480,150){\rot{0}{ \sx{3}{$v\!=\!-0.2$}} \ero} \end{picture} \end{document}