File:Aunem2ht.jpg

complex map of function AuNem with paramater $q\!=\!2$.

$u\!+\!\mathrm i v=\mathrm{AuNem}_2(x\!+\!\mathrm i y)$

The imahe is prepared for the book Superfunctions

C++ generator of map
//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=2.; 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!


 * 1) include "sune.cin"

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"

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

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; 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]);

// getchar; int M=1001,M1=M+1; int N=501,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("aunem2h.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]=   .002*(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,".003 W 0 0 0 RGB 2 setlinecap S\n"); M(-1,ti)L(1,ti) //M(-1,-ti)L(1,-ti) //M(tr,-1)L(tr,1) M(tr,0)L(tr,1) fprintf(o,".002 W 0 0 0 RGB 2 setlinecap S\n");

//M(-1,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti) //fprintf(o,".006 W 1 1 0 RGB 0 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=aune1(z); c=aunem(z); //c=sune(c); //d=sune(z); //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,".007 W 1 1 0 RGB 0 setlinecap S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf aunem2h.eps"); system( "open aunem2h.pdf"); //mac /* DO(m,50){ z=.01+.01*m; c=aune02(z); d=sune(c); p=abs(d-z)/(abs(d)+abs(z)); p=-log(p)/log(10.); c=aune0(z); d=sune(c); q=abs(d-z)/(abs(d)+abs(z)); q=-log(q)/log(10.); c=aune1(z); d=sune(c); y=abs(d-z)/(abs(d)+abs(z)); y=-log(y)/log(10.); printf("%5.3lf %5.2lf %5.2lf %5.2lf\n",Re(z), p,q,y);} return 0;}

Latex generator of labels
\documentclass{mcom-l} % \documentclass[12pt]{article} \usepackage{graphics} \paperwidth 2140pt \paperheight 1080pt \usepackage{geometry} \usepackage{rotating} \textwidth 2260pt \textheight 1260pt \topmargin -84pt \oddsidemargin -74pt \parindent 0pt \pagestyle{empty} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(2060,1052) %\put(24,20){\ing{35z}} %\put(10,10){\ing{arqnem10z}} %\put(128,50){\ing{aunem10ma}} \put(128,50){\ing{aunem2h}} \put(226,040){\sx{6.8}{\rot{0.} \bf cut \ero}} \put(1164,090){\sx{6.8}{\rot{73} \bf cut \ero}}

\put(80,1022){\sx{7.9}{$y$}} \put(30,536){\sx{7.5}{$0.5$}} \put(50,380){\sx{7.5}{$y_0$}} \put(80,036){\sx{7.5}{$0$}} %\put(-4,690){\sx{6.6}{$-y_0$}} %\put(-18, 536){\sx{6}{$-0.5$}} %\put(10, 34){\sx{6}{$-1$}} \put(80, -7){\sx{7.5}{$-1$}} \put(540,-6){\sx{7.5}{$-0.5$}} \put(1118,-6){\sx{7.5}{$0$}} \put(1190,-4){\sx{7.5}{$x_0$}} \put(1590, -6){\sx{7.5}{$0.5$}} \put(2096, -4){\sx{7.9}{$x$}} % \put(558,958){\sx{8}{\rot{-52} $u\!=\!1.2$\ero}} \put(366,710){\sx{8}{\rot{-34} $u\!=\!1.3$\ero}} \put(350,336){\sx{8}{\rot{-10} $u\!=\!1.4$\ero}} \put(760,076){\sx{8}{\rot{28} $u\!=\!1.5$\ero}} %\put(760,988){\sx{8}{\rot{-30} $u\!=\!1.5$\ero}} % \put(1072,870){\sx{8}{\rot{90} $u\!=\!1$\ero}} \put(1302,776){\sx{8}{\rot{73} $u\!=\!0.8$\ero}} \put(1840,780){\sx{8}{\rot{-5} $u\!=\!0.4$\ero}} \put(1834,686){\sx{8}{\rot{-10} $u\!=\!0.3$\ero}} \put(1834,612){\sx{8}{\rot{-18} $u\!=\!0.2$\ero}} \put(1844,534){\sx{8}{\rot{-26} $u\!=\!0.1$\ero}} \put(1844,470){\sx{8}{\rot{-32} $u\!=\!0$\ero}} \put(1844,420){\sx{8}{\rot{-48} $u\!=\!-0.1$\ero}} %\put(1844,370){\sx{8}{\rot{-62} $u\!=\!-0.2$\ero}} % \put(756,620){\sx{8}{\rot{46} $v\!=\!1.2$\ero}} \put(866,486){\sx{8}{\rot{56} $v\!=\!1.3$\ero}} % \put(1674,250){\sx{8}{\rot{65} $v\!=\!1$\ero}} \put(1820,032){\sx{8}{\rot{0} $v\!=\!0$\ero}}

%\put(1630, 864){\sx{8}{\rot{-72} $v\!=\!-1$\ero}} % %\put(1732,910){\sx{8}{\rot{88} $u\!=\!-1$\ero}} %\put(2028,714){\sx{8}{\rot{54} $u\!=\!0$\ero}} %\put(1866,386){\sx{8}{\rot{11} $u\!=\!0.3$\ero}} %\put(1860,286){\sx{8}{\rot{6} $u\!=\!0.4$\ero}} %\put(1840,180){\sx{8}{\rot{-5} $u\!=\!0.5$\ero}} \end{picture} \end{document}