Difference between revisions of "File:Nem1it.jpg"

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(miasprints)
 
(One intermediate revision by the same user not shown)
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The \( n \)th iterate of the [[Nemtsov function]] \( \mathrm{nem}_q \)
  
The \( n \)th iterate of the [[Nemtsov function]] \( \mathrm{nem}_q \)
−
is implemented thrpigh its [[superfinction]] [[SuNem]] and the [[abelfunction]] [[AuNem]]:
 +
is implemented thrpigh its [[superfunction]] [[SuNem]] and the [[abelfunction]] [[AuNem]]:
   
 
\(
  
\(
Line 41: Line 41:
 
It generates file
  
It generates file
 
nem1n6.pdf ; other files
 
nem1n6.pdf ; other files
−
nem1n6.pdf
 +
nem1n5.pdf,
−
nem1n5.pdf
 +
nem1n4.pdf,
−
nem1n3.pdf
 +
nem1n3.pdf,
−
nem1n2.pdf
 +
nem1n2.pdf,
−
nem1n1.pdf
 +
nem1n1.pdf,
−
nem1m1.pdf
 +
nem1m1.pdf,
−
nem1m2.pdf
 +
nem1m2.pdf,
−
nem1m3.pdf
 +
nem1m3.pdf,
−
nem1m4.pdf
 +
nem1m4.pdf,
−
nem1m5.pdf
 +
nem1m5.pdf,
 
nem1m6.pdf
  
nem1m6.pdf
 
are also necessary for the other pictures in this figure. They can be generated with
  
are also necessary for the other pictures in this figure. They can be generated with
−
the modifications of the code below, gradually reduceing the number of iterate from 0.6 to -0.6 ;
+
the modifications of the code below, gradually reducing the number of iterate from 0.6 to -0.6 ;
 
each time, the name of the output file needs to be corrected. No loop with respect to the number or iterate is arranged in order to keep the code simple and transparent; the number of iterate in each case is just a real constant.
  
each time, the name of the output file needs to be corrected. No loop with respect to the number or iterate is arranged in order to keep the code simple and transparent; the number of iterate in each case is just a real constant.
   
 
Then, the resulting files can be revized:
 
Then, the resulting files can be revized:
−
contours \(u=mathrm{const}\) should cross
+
contours \(u=\mathrm{const}\) should cross
−
contours \(v=mathrm{const}\) at the right angle.
+
contours \(v=\mathrm{const}\) at the right angle.
   
 
Then files nem1??.pdf can be combined with the Latex document supplied at the bottom code.
  
Then files nem1??.pdf can be combined with the Latex document supplied at the bottom code.
Line 483: Line 483:
   
 
And many other options.
  
And many other options.
  +
  +
Please, attribute the source. Please, specify your modifications, if any.
   
 
==References==
  
==References==

Latest revision as of 11:13, 6 December 2025


Figure 13.9 at page 173 of book Superfunctions[1], 2020 :

Complex maps of iterates of the Nemtsov function with parameter \(q=1\):

\(u\!+\!\mathrm i v=\mathrm{Nem}_1^{\,n}(x\!+\!\mathrm i y)\)

for \(n=0.6\), \(n=0.5\), \(n=0.4\), \(n=0.3\), \(n=0.2\), \(n=0.1\), left column, and for

\(n=-0.6\), \(n=-0.5\), \(n=-0.4\), \(n=-0.3\), \(n=-0.2\), \(n=-0.1\), right column.

The map of the identity function at \(n=0\) is not shown.

The \( n \)th iterate of the Nemtsov function \( \mathrm{nem}_q \) is implemented thrpigh its superfunction SuNem and the abelfunction AuNem:

\( \mathrm{nem}_q^n(z) = \mathrm{SuNem}_q\big(n+\mathrm{AuNem}_q(z)\big) \)

The C++ generator of the first map is loaded in the next section. It generates file nem1n6.pdf ; other files nem1n5.pdf, nem1n4.pdf, nem1n3.pdf, nem1n2.pdf, nem1n1.pdf, nem1m1.pdf, nem1m2.pdf, nem1m3.pdf, nem1m4.pdf, nem1m5.pdf, nem1m6.pdf are also necessary for the other pictures in this figure. They can be generated with the modifications of the code below, gradually reducing the number of iterate from 0.6 to -0.6 ; each time, the name of the output file needs to be corrected. No loop with respect to the number or iterate is arranged in order to keep the code simple and transparent; the number of iterate in each case is just a real constant.

Then, the resulting files can be revized: contours \(u=\mathrm{const}\) should cross contours \(v=\mathrm{const}\) at the right angle.

Then files nem1??.pdf can be combined with the Latex document supplied at the bottom code.

C++ generator of the first map

/* Files ado.cin, conto.cin, nembran.cin, arqnem.cin, sune.cin, aunemco.txt, sunemco.txt should be loaded 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"

DB Q=1.;
z_type nem(z_type z){ return z*(1.+z*z*(1.+z*Q)); }
z_type nem1(z_type z){ return 1.+z*z*(3.+z*(4.*Q)); } // WARNING: Q is global!

#include "sune.cin"
DB SUNo=0.;
z_type sunem(z_type z){ return sune(z + SUNo);}

#include"nembran.cin"
z_type NemZo=nembra(Q);
z_type ANemZo=nem(NemZo);
DB tr=Re(ANemZo);
DB ti=Im(ANemZo);

#include "arqnem.cin" 

DB C[41];
//DB K=Q*Q;
int coa(){ //C[1]=(0.5 - K)*Q; C[2]=-0.625 + (-0.25 + K/2.)*K; ..
#include "aunemco.txt"
return 40;
}

z_type aune0(z_type z){ z_type s; int n; 
s=C[40]*z; for(n=39;n>0;n--){s+=C[n];s*=z;}
return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ; 
}

DB AUNE1=0;

z_type aune1(z_type z){ int n, N=20; DO(n,N) z=arqnem(z);
return aune0(z)+(0.+n);}

z_type aunem(z_type z){ int n, N=20; DO(n,N) z=arqnem(z);
return aune0(z)+(n-AUNE1);}


z_type aune02(z_type z){ z_type s; int n; s=(C[2]*z+C[1])*z;
                return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ; }
z_type aune01(z_type z){ z_type s; int n; s=C[1]*z;
                return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) +s ; }

int main(){ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
// DB rr,ti;

K=Q*Q; x=0; co(); printf("Q=%9.4lf\n",Q);
DO(n,300) { y=Re(sunem(0)); SUNo-= y-1.; printf("%19.16lf %19.16lf\n", SUNo,y);} // getchar();

co();
coa();
AUNE1=Re(aune1(1.));
//printf("A[2][1]= %9.5f\n",A[2][1]);
printf("A[2][0]= %9.5lf , A[2][1]=%9.5lf\n",A[2][0],A[2][1]);
printf("A[3][0]= %9.5lf , A[3][1]=%9.5lf\n",A[3][0],A[3][1]);

printf("K= %9.5lf , C[1]=%9.5lf\n",K,C[1]);

 int M=1001,M1=M+1;
 int N=201,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
//DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
FILE *o;o=fopen("nem1n6.eps","w");ado(o,2008,1008);
fprintf(o,"1004 4 translate\n 1000 1000 scale 2 setlinecap\n");
DO(m,M1) X[m]=-1.+.002*(m-.5);
DO(n,N1) Y[n]=    .005*(n-.5);
for(m=-10;m<11;m+=5){ M(.1*m,0)L(.1*m,1)}
for(n=0;n<11;n+=5){ M(-1,.1*n)L(1,.1*n)}
fprintf(o,".004 W 0 0 0 RGB 2 setlinecap S\n");
M(-1,ti)L(1,ti) M(tr,0)L(tr,1) fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");

DO(m,M1)DO(n,N1){g[m*N1+n]=9999999; f[m*N1+n]=9999999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
c=aunem(z); c=sunem(.6+c);
//p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); 
       p=Re(c); q=Im(c);
if(p>-85 && p<85) g[m*N1+n]=p;
        if(p>-1001 && p<1001 && 
        q >-1001 && q<1001 ) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

//M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti)
//fprintf(o,".002 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=200.;q=2.;
//#include"plofu.cin"

for(m=-5;m<5;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".001 W 0 .6 0 RGB S\n");
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".001 W .9 0 0 RGB S\n");
for(m=0;m<5;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".001 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,".004 W .8 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,".004 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".004 W .5 0 .5 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".004 W 0 0 0 RGB S\n");

/*
//conto(o,g,w,v,X,Y,M,N,15.5,-1,1);fprintf(o,".02 W 1 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,15.,-p,p);fprintf(o,".004 W 0 0 1 RGB S\n");
conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".002 W 0 1 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".002 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,12.,-p,p);fprintf(o,".004 W 0 0 .7 RGB S\n");
conto(o,g,w,v,X,Y,M,N,11.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N,10.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 9.,-p,p);fprintf(o,".004 W 0 .6 .8 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 8.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 7.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 6.,-p,p);fprintf(o,".004 W 0 .6 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 5.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 4.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 3.,-p,p);fprintf(o,".004 W 1 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 2.,-p,p);fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,g,w,v,X,Y,M,N, 1.,-p,p);fprintf(o,".005 W .5 0 0 RGB S\n");
*/

M(-1,0) L(0,0) M(tr,ti)L(0,0) L(tr,-ti)
fprintf(o,".008 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf nem1n6.eps");
        system( "open nem1n6.pdf"); //mac
return 0;}

Latex combiner of the maps and labels

\documentclass{mcom-l}
% \documentclass[12pt]{article}
 \usepackage{graphics}
 \paperwidth 366pt
 \paperheight 566pt
 \usepackage{geometry}
 \usepackage{rotating}
 \textwidth 660pt
 \textheight 660pt
 \topmargin -96pt
 \oddsidemargin -220pt
 \pagestyle{empty}
%\parindent 0pt
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
%\pagestyle{empty}
\newcommand \scalite {\put(200,040){\sx{8}{\rot{0.} \bf cut \ero}}
\put(1130,090){\sx{8}{\rot{73} \bf cut \ero}}
\put(50,1022){\sx{9}{$y$}}
\put(-12,536){\sx{9}{$0.5$}}
\put(20,380){\sx{9}{$y_0$}}
\put(50,030){\sx{9}{$0$}}
\put(60, -9){\sx{9}{$-1$}}
\put(510,-9){\sx{9}{$-0.5$}}
\put(1084,-9){\sx{9}{$0$}}
\put(1150,-7){\sx{9}{$x_0$}}
\put(1560, -9){\sx{9}{$0.5$}}
\put(2076, -9){\sx{9}{$x$}}
}
%\url{http://mizugadro.mydns.jp/t/index.php/File:Aunem2ht.jpg}\\
\begin{figure} 

\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1n6}}
%\put(770,980){\sx{9}{\rot{-52} $u\!=\!0$\ero}}
\put(110,880){\sx{9}{\rot{-88} $u\!=\!-2$\ero}}
\put(412,880){\sx{9}{\rot{-80} $u\!=\!-1$\ero}}
\put(1270,594){\sx{9}{\rot{17} $u\!=\!0$\ero}}
\put(1850,110){\sx{9}{\rot{77} $u\!=\!1$\ero}}
%
\put(150,880){\sx{9}{\rot{14} $v\!=\!-1$\ero}}
\put(740,640){\sx{9}{\rot{20} $v\!=\!0$\ero}}
\put(140,290){\sx{9}{\rot{-20} $v\!=\!1$\ero}}
%
\put(1940,914){\sx{9}{\rot{-49} $v\!=\!2$\ero}}
\put(1810,534){\sx{9}{\rot{-31} $v\!=\!1$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}} 
\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1m6}}
\put(680,920){\sx{9}{\rot{-17} $u\!=\!0.4$\ero}}
\put(650,700){\sx{9}{\rot{-7} $u\!=\!0.3$\ero}}
\put(650,486){\sx{9}{\rot{2} $u\!=\!0.2$\ero}}
\put(650,280){\sx{9}{\rot{14} $u\!=\!0.1$\ero}}
\put(730,100){\sx{9}{\rot{23} $u\!=\!0$\ero}}
%
\put(374,770){\sx{9}{\rot{85} $v\!=\!1$\ero}}
\put(1226,800){\sx{9}{\rot{50} $v\!=\!0.6$\ero}}
\put(1420,756){\sx{9}{\rot{40} $v\!=\!0.5$\ero}}
\put(1580,660){\sx{9}{\rot{30} $v\!=\!0.4$\ero}}
\put(1690,530){\sx{9}{\rot{22} $v\!=\!0.3$\ero}}
\put(1760,370){\sx{9}{\rot{14} $v\!=\!0.2$\ero}}
\put(1790,204){\sx{9}{\rot{6} $v\!=\!0.1$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}}

\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1n5}}
%\put(640,814){\sx{9}{\rot{-33} $u\!=\!0$\ero}}
\put(1270,596){\sx{9}{\rot{30} $u\!=\!0$\ero}}
\put(244,600){\sx{9}{\rot{-59} $u\!=\!1$\ero}}
\put(1886,110){\sx{9}{\rot{79} $u\!=\!1$\ero}}
%
\put(260,710){\sx{9}{\rot{43} $v\!=\!-1$\ero}}
\put(918,740){\sx{9}{\rot{72} $v\!=\!0$\ero}}
%
\put(1940,1030){\sx{9}{\rot{-41} $v\!=\!2$\ero}}
\put(1670,700){\sx{9}{\rot{-36} $v\!=\!1$\ero}}
%
\put(1810,32){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}} 
\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1m5}}
\put(850,1024){\sx{9}{\rot{-32} $u\!=\!0.4$\ero}}
\put(750,860){\sx{9}{\rot{-26} $u\!=\!0.3$\ero}}
\put(700,674){\sx{9}{\rot{-18} $u\!=\!0.2$\ero}}
\put(670,500){\sx{9}{\rot{-10} $u\!=\!0.1$\ero}}
\put(670,314){\sx{9}{\rot{1} $u\!=\!0$\ero}}
\put(670,130){\sx{9}{\rot{10} $u\!=\!-0.1$\ero}}
%
\put(280,800){\sx{9}{\rot{75} $v\!=\!1.1$\ero}}
\put(500,770){\sx{9}{\rot{73} $v\!=\!1$\ero}}
\put(1310,790){\sx{9}{\rot{42} $v\!=\!0.6$\ero}}
\put(1480,730){\sx{9}{\rot{33} $v\!=\!0.5$\ero}}
\put(1590,610){\sx{9}{\rot{26} $v\!=\!0.4$\ero}}
\put(1700,490){\sx{9}{\rot{20} $v\!=\!0.3$\ero}}
\put(1770,340){\sx{9}{\rot{13} $v\!=\!0.2$\ero}}
\put(1790,190){\sx{9}{\rot{5} $v\!=\!0.1$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}}

\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1n4}}
\put(1250,600){\sx{9}{\rot{40} $u\!=\!0$\ero}}
\put(1926,110){\sx{9}{\rot{82} $u\!=\!1$\ero}}
\put(126,230){\sx{9}{\rot{-30} $u\!=\!1$\ero}}

\put(284,710){\sx{9}{\rot{80} $v\!=\!-1$\ero}}
\put(840,700){\sx{9}{\rot{90} $v\!=\!0$\ero}}
%
%\put(1940,1030){\sx{9}{\rot{-41} $v\!=\!2$\ero}}
\put(1670,748){\sx{9}{\rot{-30} $v\!=\!1$\ero}}
%
\put(1810,32){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}} 
\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1m4}}
\put(870,970){\sx{9}{\rot{-42} $u\!=\!0.3$\ero}}
\put(780,830){\sx{9}{\rot{-36} $u\!=\!0.2$\ero}}
\put(720,676){\sx{9}{\rot{-32} $u\!=\!0.1$\ero}}
\put(700,510){\sx{9}{\rot{-23} $u\!=\!0$\ero}}
\put(690,340){\sx{9}{\rot{-13} $u\!=\!-0.1$\ero}}
\put(680,180){\sx{9}{\rot{-4} $u\!=\!-0.2$\ero}}
%
\put(344,720){\sx{9}{\rot{65} $v\!=\!1.1$\ero}}
\put(530,680){\sx{9}{\rot{64} $v\!=\!1$\ero}}
\put(1320,734){\sx{9}{\rot{33} $v\!=\!0.6$\ero}}
\put(1480,672){\sx{9}{\rot{25} $v\!=\!0.5$\ero}}
\put(1590,566){\sx{9}{\rot{19} $v\!=\!0.4$\ero}}
\put(1700,452){\sx{9}{\rot{14} $v\!=\!0.3$\ero}}
\put(1770,320){\sx{9}{\rot{10} $v\!=\!0.2$\ero}}
\put(1790,174){\sx{9}{\rot{5} $v\!=\!0.1$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}}

\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1n3}}
\put(1240,630){\sx{9}{\rot{50} $u\!=\!0$\ero}}
\put(1970,110){\sx{9}{\rot{86} $u\!=\!1$\ero}}
%
\put(1670,800){\sx{9}{\rot{-22} $v\!=\!1$\ero}}
\put(660,750){\sx{9}{\rot{-70} $v\!=\!0$\ero}}
\put(226,894){\sx{9}{\rot{22} $u\!=\!1$\ero}}
%
%
\put(130,270){\sx{9}{\rot{-88} $v\!=\!1$\ero}}
\end{picture}} 
\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1m3}}
\put(880,960){\sx{9}{\rot{-50} $u\!=\!0.2$\ero}}
\put(810,820){\sx{9}{\rot{-46} $u\!=\!0.1$\ero}}
\put(734,690){\sx{9}{\rot{-42} $u\!=\!0$\ero}}
\put(650,580){\sx{9}{\rot{-39} $u\!=\!-0.1$\ero}}
\put(560,470){\sx{9}{\rot{-35} $u\!=\!-0.2$\ero}}
\put(490,330){\sx{9}{\rot{-31} $u\!=\!-0.3$\ero}}
\put(420,200){\sx{9}{\rot{-28} $u\!=\!-0.4$\ero}}
%
\put(342,720){\sx{9}{\rot{52} $v\!=\!1.1$\ero}}
\put(500,666){\sx{9}{\rot{50} $v\!=\!1$\ero}}
\put(1340,696){\sx{9}{\rot{25} $v\!=\!0.6$\ero}}
\put(1490,630){\sx{9}{\rot{18} $v\!=\!0.5$\ero}}
\put(1590,524){\sx{9}{\rot{15} $v\!=\!0.4$\ero}}
\put(1700,420){\sx{9}{\rot{10} $v\!=\!0.3$\ero}}
\put(1770,300){\sx{9}{\rot{5} $v\!=\!0.2$\ero}}
\put(1790,166){\sx{9}{\rot{2} $v\!=\!0.1$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}}

\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1n2}}
\put(338,840){\sx{9}{\rot{45} $u\!=\!-1$\ero}}
\put(1240,686){\sx{9}{\rot{65} $u\!=\!0$\ero}}
\put(2018,110){\sx{9}{\rot{87} $u\!=\!1$\ero}}
%
\put(1660,870){\sx{9.4}{\rot{-16} $v\!=\!1$\ero}}
\put(660,440){\sx{9}{\rot{-52} $v\!=\!0$\ero}}
\put(1800,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}} 
\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1m2}}
\put(1060,940){\sx{9}{\rot{-65} $u\!=\!0.2$\ero}}
\put(950,870){\sx{9}{\rot{-62} $u\!=\!0.1$\ero}}
\put(850,800){\sx{9}{\rot{-60} $u\!=\!0$\ero}}
\put(750,730){\sx{9}{\rot{-57} $u\!=\!-0.1$\ero}}
\put(650,660){\sx{9}{\rot{-55} $u\!=\!-0.2$\ero}}
\put(540,560){\sx{9}{\rot{-52} $u\!=\!-0.3$\ero}}
\put(440,480){\sx{9}{\rot{-49} $u\!=\!-0.4$\ero}}
\put(340,390){\sx{9}{\rot{-48} $u\!=\!-0.5$\ero}}
%
\put(220,736){\sx{9}{\rot{34} $v\!=\!1.1$\ero}}
\put(370,670){\sx{9}{\rot{34} $v\!=\!1$\ero}}
\put(1280,900){\sx{9}{\rot{20} $v\!=\!0.8$\ero}}
\put(1420,694){\sx{9}{\rot{14} $v\!=\!0.6$\ero}}
\put(1540,602){\sx{9}{\rot{12} $v\!=\!0.5$\ero}}
\put(1620,500){\sx{9}{\rot{8} $v\!=\!0.4$\ero}}
\put(1700,390){\sx{9}{\rot{8} $v\!=\!0.3$\ero}}
\put(1770,276){\sx{9}{\rot{4} $v\!=\!0.2$\ero}}
\put(1790,154){\sx{9}{\rot{2} $v\!=\!0.1$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
\end{picture}}

\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1n1}}
\put(244,700){\sx{9}{\rot{70} $u\!=\!-1$\ero}}
\put(1225,840){\sx{9}{\rot{76} $u\!=\!0$\ero}}
\put(2070,110){\sx{9}{\rot{88} $u\!=\!1$\ero}}
%
\put(1810,924){\sx{9}{\rot{-9} $v\!=\!1$\ero}}
\put(660,244){\sx{9}{\rot{-26} $v\!=\!0$\ero}}
\put(1810,34){\sx{9}{\rot{0} $v\!=\!0$\ero}}
%
\end{picture}} 
\sx{.085}{\begin{picture}(2100,1100)\scalite \put(100,50){\ing{nem1m1}}
%
\put(1178,960){\sx{9}{\rot{-79} $u\!=\!0.2$\ero}}
\put(1060,960){\sx{9}{\rot{-79} $u\!=\!0.1$\ero}}
\put(964,896){\sx{9}{\rot{-76} $u\!=\!0$\ero}}
\put(850,870){\sx{9}{\rot{-73} $u\!=\!-0.1$\ero}}
\put(740,830){\sx{9}{\rot{-72} $u\!=\!-0.2$\ero}}
\put(630,780){\sx{9}{\rot{-71} $u\!=\!-0.3$\ero}}
\put(530,720){\sx{9}{\rot{-70} $u\!=\!-0.4$\ero}}
\put(420,680){\sx{9}{\rot{-69} $u\!=\!-0.5$\ero}}
\put(320,640){\sx{9}{\rot{-69} $u\!=\!-0.6$\ero}}
\put(210,590){\sx{9}{\rot{-69} $u\!=\!-0.7$\ero}}
\put(110,500){\sx{9}{\rot{-68} $u\!=\!-0.8$\ero}}
%
\put(220,920){\sx{9}{\rot{19} $v\!=\!1.1$\ero}}
\put(370,840){\sx{9}{\rot{19} $v\!=\!1$\ero}}
%\put(1200,956){\sx{9}{\rot{10} $v\!=\!0.9$\ero}}
\put(1350,870){\sx{9}{\rot{8} $v\!=\!0.8$\ero}}
\put(1490,670){\sx{9}{\rot{7} $v\!=\!0.6$\ero}}
\put(1580,570){\sx{9}{\rot{5} $v\!=\!0.5$\ero}}
\put(1650,466){\sx{9}{\rot{3} $v\!=\!0.4$\ero}}
\put(1730,358){\sx{9}{\rot{3} $v\!=\!0.3$\ero}}
\put(1790,254){\sx{9}{\rot{2} $v\!=\!0.2$\ero}}
\put(1800,144){\sx{9}{\rot{1} $v\!=\!0.1$\ero}}
\put(1810,33){\sx{9}{\rot{0} $v\!=\!0$\ero}}
%
\end{picture}}
\vskip -6pt
\caption{ $u\!+\!\mathrm i v=\mathrm{Nem}_1^{\,n}(x\!+\!\mathrm i y)$ for $n=-0.6$ .. $0.6$}
\end{figure}
\end{document}

Warning

In the C++ code above, the abelfuction \( \mathrm{AuNem}_q \) is implemented as routine z_type aunem(z_type z); parameter \(q\) is stored in the global variable Q.

Sorry for the invonvenience: First, the pictures are plotted; then, it is possible to understand which notations are better to use in the code.

Colleagues are cordially invited to modify the code:

You may modify the names of the variables to something more convenient. You may try to avoid use of the global variables Q, tr, ti. You may add your internal tests.

You may use the contour drawing Conrec6.cin (written by ChatGPT) (instead of old routine conto.cin) to plot the Complex maps.
Take into accoint that the call of routine "conrec6" is different from call of routine "conto"; use Conrec6 demo as an example of the calling.

You may extact routine aunem into a separate file, as an excersice.

You may arrange the loop with respect to the numbner of iterate.

You may play with parameter \(q\) of the Nemtsov function (in this code, \(q=1\)).

And many other options.

Please, attribute the source. Please, specify your modifications, if any.

References

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