File:Logi2qt.jpg

Fig. 7.13 from book Superfunctions.

Complex map of function \( T \) defined with

\( T(z)=-1.8\,(-z)^{\sqrt{2}} \)

The map is constructed with lines

\( u= \Re(T(x+\mathrm i y ))= \rm const \)

and lines

\( v= \Im(T(x+\mathrm i y ))= \rm const \)

in the \(x,y\) plane.

C++ generator of map

//You need to load ado.cin and conto.cin in order 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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
//#include "efjh.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=101,M1=M+1;
  int N=201,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("logi2b3.eps","w");ado(o,124,124);
FILE *o;o=fopen("logi2q.eps","w");ado(o,124,124);
fprintf(o,"62 62 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-3.+.06*(m-.5);
DO(n,N1) Y[n]=-3.+.03*(n-.5);

for(m=-3;m<4;m++){if(m==0){M(m,-3.04)L(m,3.04)} else{M(m,-3)L(m,3)}}
for(n=-3;n<4;n++){ M( -3 ,n)L(3,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");

// maq(3.);
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=F(.5+E(z));
        c=-1.8*exp(sqrt(2.)*log(-z));
        p=Re(c);q=Im(c);
        if(p>-4.9 && p<4.9) {g[m*N1+n]=p;}
        if(q>-4.9 && q<4.9 && fabs(q)>1.e-11 ) {f[m*N1+n]=q;}
                        }}

fprintf(o,"1 setlinejoin 2 setlinecap\n");
//p=.8;q=.4;
p=2.;q=.5;
//#include"plof.cin"
for(m=-2;m<2;m++)
for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 W 0 .6 0 RGB S\n");
for(m=0;m<2;m++)
for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".005 W .9 0 0 RGB S\n");
for(m=0;m<2;m++)
for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 W 0 0 .9 RGB S\n");

for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".02 W .9 0 0 RGB S\n");
for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 .9 RGB S\n");
for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 W 0 0 0 RGB S\n");

                  conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 W .6 0 .6 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n");

//M(-3.02,0)L(0,0)
//M(1.-1./Q,0)L(3,0) fprintf(o,"0.03 W 1 1 1 RGB S\n");
M(0,0)L(3,0) fprintf(o,"0.03 W 1 1 1 RGB S\n");
//for(n=0;n<16;n++) {M(-.2*n+.4,0)L(-.2*(n+.8),0)}
for(n=0;n<15;n++) { M(.2*(n+.4),0)
                        L(.2*(n+.8),0)}
fprintf(o,"0.04 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi2q.eps"); // both linux and mac
        system( "xpdf logi2q.pdf"); // for linux
//	system( "open logi2q.pdf"); // for mac
//	getchar(); system("killall Preview"); for mac
}

//

Latex generator of labels

% file logi2q.jpg should be generated with the C++ code above in order to latex the code below:

%

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 640pt
\paperheight 640pt
\textwidth 700pt
\textheight 700pt
\topmargin -92pt
\oddsidemargin -91pt
\newcommand \sx {\scalebox}
\newcommand \ing \includegraphics
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}

\newcommand \axesS {
\normalsize
\put(  18,624){\sx{2.6}{$y$}}
\put(  18,530){\sx{2.6}{$2$}}
\put(  18,430){\sx{2.6}{$1$}}
\put(  18,330){\sx{2.6}{$0$}}
\put(  -2,230){\sx{2.6}{$-1$}}
\put(  -2,130){\sx{2.6}{$-2$}}
\put(  -2, 30){\sx{2.6}{$-3$}}
\put(  16, 10){\sx{2.6}{$-3$}}
\put(116, 10){\sx{2.6}{$-2$}}
\put(216, 10){\sx{2.6}{$-1$}}
\put(332, 10){\sx{2.6}{$0$}}
\put(432, 10){\sx{2.6}{$1$}}
\put(532, 10){\sx{2.6}{$2$}}
\put(622, 10){\sx{2.6}{$x$}}
}
\begin{picture}(640,628) 
%\put( 4, 4){\ing{logi2b3}}
%\put(36, 36){\ing{logi2s5}}
%\put(36, 36){\ing{logi2q}}
\put(28, 28){\sx{5}{\ing{logi2q}}}
\normalsize
\put( 52, 430){\rot{ 8}\sx{4}{$v\!=\!4$}\ero}
\put( 60, 326){\rot{ 0}\sx{4}{$v\!=\!0$}\ero}
\put( 566, 328){\rot{ 0}\sx{4}{\bf cut}\ero}
%\put( 42, 232){\rot{-9}\sx{4}{$v\!=\!-4$}\ero}
%
\put( 86, 42){\rot{71}\sx{4}{$u\!=\!-4$}\ero}
\put( 128, 42){\rot{68}\sx{4}{$u\!=\!-3$}\ero}
\put( 208, 44){\rot{64}\sx{4}{$u\!=\!0$}\ero}
\put( 288, 39){\rot{56}\sx{4}{$u\!=\!3$}\ero}
%
\put( 430, 532){\rot{ 48}\sx{4}{$v\!=\!2$}\ero}
\put( 468, 488){\rot{ 52}\sx{4}{$v\!=\!0$}\ero}
\put( 514, 454){\rot{ 56}\sx{4}{$v\!=\!-2$}\ero}%%
\put( 564, 424){\rot{ 60}\sx{4}{$v\!=\!-4$}\ero}

\put( 564, 204){\rot{-60}\sx{4}{$v\!=\!4$}\ero}
\put( 524, 170){\rot{-58}\sx{4}{$v\!=\!2$}\ero}
\put( 470, 150){\rot{-54}\sx{4}{$v\!=\!0$}\ero}
\put( 410, 132){\rot{-50}\sx{4}{$v\!=\!-2$}\ero}
\axesS
\end{picture}
\end{document}

%

References

https://mizugadro.mydns.jp/BOOK/463.pdf Dmitrii Kouznetsov. Superfunctions. 2020 July.