File:Arqnem10zt.jpg

Complex map of function ArqNem with parameter $q\!=\!1$:

$u\!+\!\mathrm i v = \mathrm{ArqNem}_1(x\!+\!\mathrm i y)$.

Additional grid lines mark the branch point of function $\mathrm{arqNem}_q$;

$x_0 +\mathrm i y_0 = \mathrm{Nem}_1(z_0)$

where $z_0$ is solution of equation $\mathrm{Nem}_1^{~\prime} (z_0)=0$

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

//using namespace std; typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) 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!

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


 * 1) include "arqnem.cin"

/* z_type arnemU(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q; z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b); z_type R=-I*sqrt(-r); z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.); z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S); z_type h=0.25/q2 + B; z_type H=I*sqrt(-h); z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H); z_type G=I*sqrt(-g); return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arnemD(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q; z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b); z_type R=I*sqrt(-r); z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.); z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S); z_type h=0.25/q2 + B; z_type H=-I*sqrt(-h); z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H); z_type G=-I*sqrt(-g); return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arnemR(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q; z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b); z_type R=sqrt(r); z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.); z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S); z_type h=0.25/q2 + B; z_type H=sqrt(h); z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H); z_type G=sqrt(g); return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arnemL(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q; z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b); z_type R=-sqrt(r); z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.); z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S); z_type h=0.25/q2 + B; z_type H=sqrt(h); z_type g=0.5/q2 - B + (.25+2.*q2)/(q3*H); z_type G=sqrt(g); return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arqnem(z_type z){ DB x,y; x=Re(z);y=Im(z); if( y>ti || (x<0 && y>=0)) return arnemU(z); //if(y<0) return conj(arnemU(conj(z))); if(y<-ti || (x<0 && y<=0)) return arnemD(z); if(x*ti>fabs(y)*tr) return arnemR(z); return arnemL(z); }

int main{ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; // DB rr,ti; int M=2001,M1=M+1; int N=1001,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. //char v[M1*N1]; // v is working array FILE *o;o=fopen("arqnem10z.eps","w");ado(o,2020,2020); fprintf(o,"1010 1010 translate\n 1000 1000 scale 2 setlinecap\n"); DO(m,M1) X[m]=-1.+.001*(m-.5); DO(n,N1) Y[n]=-1.+.002*(n-.5); for(m=-10;m<11;m+=5){ M(.1*m,-1)L(.1*m,1)} for(n=-10;n<11;n+=5){ M(-1,.1*n)L(1,.1*n)} fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");

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

M(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti) fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n"); printf("ti, tr= %9.5lf %9.5lf\n",tr,ti);

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=sunem(z); //c=aun(z); c=arqnem(z); //c=arnemD(z); //c=nem(c); //p=abs(c-z)/abs(c+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=20.;q=.5; //#include"plofu.cin" //p=2;q=1;

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,".006 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,".006 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,".006 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,".01 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,".01 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".01 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,".01 W 0 0 0 RGB S\n");

/* // conto(o,g,w,v,X,Y,M,N,15.5,-p,p);fprintf(o,".02 W .3 0 .3 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(-2,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti) fprintf(o,".004 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf arqnem10z.eps"); system(   "open arqnem10z.pdf"); //mac

//getchar; system("killall Preview");// mac

return 0; }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphics} \paperwidth 2020pt \paperheight 2020pt \usepackage{geometry} \usepackage{rotating} \textwidth 2260pt \textheight 2260pt \topmargin -97pt \oddsidemargin -84pt \parindent 0pt \pagestyle{empty} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(2020,2020) %\put(24,20){\ing{35z}} \put(10,10){\ing{arqnem10z}} \put(226,1000){\sx{8}{\rot{0.} \bf cut \ero}} \put(1044,1044){\sx{8}{\rot{73} \bf cut \ero}}

\put(940,1980){\sx{8}{$y$}} \put(900,1500){\sx{8}{$0.5$}} \put(900,1354){\sx{8}{$y_0$}} \put(930,1000){\sx{8}{$0$}} \put(860,664){\sx{8}{$-y_0$}} \put(830, 492){\sx{8}{$-0.5$}} \put(400, 942){\sx{8}{$-0.5$}} \put(1000, 942){\sx{8}{$0$}} \put(1064, 942){\sx{8}{$x_0$}} \put(1464, 942){\sx{8}{$0.5$}} \put(1980, 942){\sx{8}{$x$}} % \put(700,1820){\sx{8}{\rot{-70} $v\!=\!0.6$\ero}} \put(470,1540){\sx{8}{\rot{-46} $v\!=\!0.7$\ero}} \put(240,1220){\sx{8}{\rot{-31} $v\!=\!0.8$\ero}} % \put(1048,1670){\sx{8}{\rot{90} $v\!=\!0.5$\ero}} \put(1244,1644){\sx{8}{\rot{66} $v\!=\!0.4$\ero}} \put(1420,1544){\sx{8}{\rot{49} $v\!=\!0.3$\ero}} \put(1560,1400){\sx{8}{\rot{32} $v\!=\!0.2$\ero}} \put(1620,1206){\sx{8}{\rot{14} $v\!=\!0.1$\ero}} \put(1620,1000){\sx{8}{\rot{0} $v\!=\!0$\ero}} \put(1600, 798){\sx{8}{\rot{-16} $v\!=\!-0.1$\ero}} % \put(1210, 740){\sx{8}{\rot{77} $u\!=\!0.2$\ero}} \put(1280, 667){\sx{8}{\rot{65} $u\!=\!0.3$\ero}} \put(1380, 580){\sx{8}{\rot{57} $u\!=\!0.4$\ero}} \put(1500, 430){\sx{8}{\rot{51} $u\!=\!0.5$\ero}} \put(1654, 220){\sx{8}{\rot{48} $u\!=\!0.6$\ero}}

\end{picture} \end{document}