File:Nem1hht.jpg

Range of validity of relation

$h(h(x\!+\!\mathrm i y))=\mathrm{Nem}_1(x\!+\!\mathrm i y)$

is shaded with rectangular grid; here $h$ is iterate half of the Nemtsov function with parameter $q\!=\!1$,

$h(z)=\mathrm{Nem}_1^{1/2}(z)=\mathrm{SuNem}_1\!\left( \frac{1}{2} + \mathrm{AuNem}_1(z) \right)$

is expressed through functions SuNem and AuNem.

The cut lines of function ArqNem are marked with yellow. Additional grid lines $x\!=\!x_0$ and $y\!=\!y_0$ indicate value $z_0=x_0\!+\!\mathrm i y_0=\mathrm{NemBran}(1)$ which is branch point of function $\mathrm{ArcNem}_1$. It is expressed through function NemBran.

This image is used as figure 12 in the preliminary version of the article Nemtsov function and its iterates

Also, this image is planned to include into the English version of book Superfunctions

C++ generator of the cut lines
//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!

DB SUNo=0.; z_type sunem(z_type z){ return sune(z + SUNo);}
 * 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;

K=Q*Q; x=0; co; printf("Q=%9.4lf\n",Q); DO(n,30) { 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]);

// getchar; int M=501,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("nem1hha.eps","w");ado(o,2008,2008); fprintf(o,"1004 4 translate\n 1000 1000 scale 2 setlinecap\n"); DO(m,M1) X[m]=-.5+.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,2)} for(n=0;n<21;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(-1,-ti)L(1,-ti) //M(tr,-1)L(tr,1) M(tr,0)L(tr,1) fprintf(o,".003 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=sunem(.5+c); c=aunem(c); c=sunem(.5+c); d=nem(z); p=abs(c-d)/(abs(c)+abs(d)); 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,".01 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 nem1hha.eps"); system( "open   nem1hha.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;}

C++ generator of the shading
//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!

DB SUNo=0.; z_type sunem(z_type z){ return sune(z + SUNo);}
 * 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;

K=Q*Q; x=0; co; printf("Q=%9.4lf\n",Q); DO(n,30) { 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]);

// getchar; int M=501,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("aunem1h.eps","w");ado(o,2008,1008); FILE *o;o=fopen("nem1hhb.eps","w");ado(o,2008,2008); fprintf(o,"1004 4 translate\n 1000 1000 scale 2 setlinecap\n"); DO(m,M1) X[m]=-1.+.004*(m-.5); DO(n,N1) Y[n]=   .004*(n-.5); for(m=-10;m<11;m+=5){ M(.1*m,0)L(.1*m,2)} for(n=0;n<21;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(-1,-ti)L(1,-ti) //M(tr,-1)L(tr,1) M(tr,0)L(tr,1) fprintf(o,".003 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=sunem(.5+c); c=aunem(c); c=sunem(.5+c); //c=arqnem(c); d=nem(z); p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.); //      p=Re(z); q=Im(z); //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; } if(p>10) { g[m*N1+n]=Re(z); f[m*N1+n]=Im(z);} }}

//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,".004 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,".004 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,".004 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,".008 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,".008 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".008 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,".008 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 nem1hhb.eps"); system( "open nem1hhb.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 the labels
\documentclass{mcom-l} % \documentclass[12pt]{article} \usepackage{graphics} \paperwidth 366pt \paperheight 360pt \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(50,2022){\sx{9}{$y$}} \put(-12,1536){\sx{9}{$1.5$}} \put(50,1036){\sx{9}{$1$}} \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}} \sx{.17}{\begin{picture}(2100,2100) %\scalite \put(50,2022){\sx{9}{$y$}} \put(-12,1536){\sx{9}{$0.5$}} \put(50,1036){\sx{9}{$1$}} \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}{$x$}} %\put(100,50){\ing{71}} %\put(100,50){\ing{72}} \put(100,50){\ing{nem1hha}} \put(100,50){\ing{nem1hhb}} \end{picture}} \end{figure} \end{document}