File:Sunem0ht.jpg

Complex map of function SuNem with parameter $q\!=\!0$:

$u\!+\!\mathrm i v=\mathrm{SuNem}_{0}(x\!+\!\mathrm i y)$

SuNem is superfunction for the Nemtsov function,

$\mathrm{nem}(z)=z+z^3+qz^4$

The map is symmetric with respect to reflection from the real axis; so, the only upper half of the complex plane is shown.

Usage
This image is intended to be used as top picture of figure 7 (label "sums") of article Nemtsov function and its iterates

Then, it should be included in to the English version of the book Superfunctions

Mathematica generator of the algorithm
T[z_] = z + z^3 + q z^4 P[m_, L_] := Sum[a[m, n] L^n, {n, 0, IntegerPart[m/2]}] A[1, 0] = -q; A[1, 1] = 0; a[2, 0] = 0; A[2, 0] = 0; F[m_,z_]:=1/(-2z)^(1/2)(1-q/(-2z)^(1/2)+ Sum[P[n,Log[-z]]/(-2z)^(n/2),{n,2,m}])

m = 2; s[m] =Numerator[Normal[Series[ (T[F[m,-1/x^2]]-F[m,-1/x^2+1])2^((m+1)/2)/x^(m+2),{x,0,1}]]] t[m] = Numerator[Coefficient[Normal[s[m]], x] ] sub[m] = Extract[Solve[t[m] == 0, a[m, 1]], 1] SUB[m] = Simplify[sub[m]] f[m, z_] = ReplaceAll[F[m, z], SUB[m]]

m = 3 s[m] =Simplify[ReplaceAll[Series[ (T[F[m,-1/x^2]]-F[m,-1/x^2+1])2^((m+3)/2)/x^(m+3),{x,0,0}],SUB[m-1]]]; t[m] = ReplaceAll[Normal[s[m]], Log[x] -> L]; u[m] = Table[Coefficient[t[m] L, L^n] == 0, {n,1,1+IntegerPart[m/2]}]; tab[m] = Table[a[m, n], {n, 0, IntegerPart[m/2]}]; sub[m] = Extract[Solve[u[m], tab[m]], 1] SUB[m] = Join[SUB[m-1], sub[m]];

(* and so on for m=4,  m=5, etc. *)

C++ generator of the map
Files ado.cin, conto.cin, sune.cin, sunema.txt should be loaded for the compilation of 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=0.; 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"

DB SUNo=0; // should be reassigned z_type sunem(z_type z){ return sune(z + SUNo);}

int main{ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; // DB rr,ti; co; // Evaluate soefficients of the expansion

x=0; DO(n,60) { y=Re(sune(x)); x-= y-1.; printf("%19.16lf %19.16lf\n", x,y);} SUNo=x; 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]); int M=501,M1=M+1; int N=251,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("sunem0h.eps","w");ado(o,1008,508); fprintf(o,"504 4 translate\n 100 100 scale 2 setlinecap\n"); DO(m,M1) X[m]=-5.+.02*(m-.5); DO(n,N1) Y[n]=0.+.02*(n-.5); for(m=-5;m<6;m+=1){ M(m,0)L(m,5)} for(n=0;n<6;n+=1){ M(-5,n)L(5,n)} 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=sunem(z); //d=sune(z); //p=abs(c-d)/abs(c+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=6.;q=.6; //#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,".01 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,".01 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,".01 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,".023 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,".023 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".023 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,".023 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,".04 W 0 0 1 RGB S\n"); conto(o,g,w,v,X,Y,M,N,14.,-p,p);fprintf(o,".02 W 0 1 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,13.,-p,p);fprintf(o,".02 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,12.,-p,p);fprintf(o,".04 W 0 0 .7 RGB S\n"); conto(o,g,w,v,X,Y,M,N,11.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,10.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,9.,-p,p);fprintf(o,".04 W 0 .6 .8 RGB S\n"); conto(o,g,w,v,X,Y,M,N,8.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,7.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,6.,-p,p);fprintf(o,".04 W 0 .6 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,5.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,4.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,3.,-p,p);fprintf(o,".04 W 1 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,2.,-p,p);fprintf(o,".02 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,1.,-p,p);fprintf(o,".05 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 sunem0h.eps"); system(   "open sunem0h.pdf"); //mac

//for(m=-40;m<0;m++) {x=m; y=Re(sun(x)); t=Re(nem(sun(x-1.))); printf("%6.1lf %18.15lf %18.15lf %18.15lf\n", x,y,t,y-t);} return 0; }

Latex generator of the labels
%\documentclass[12pt]{article} \documentclass{mcom-l} \usepackage{graphics} \paperwidth 1030pt \paperheight 530pt \usepackage{geometry} \usepackage{rotating} \textwidth 1260pt \textheight 1260pt \topmargin -96pt \oddsidemargin -82pt \parindent 0pt \pagestyle{empty} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(1038,530) %\put(50,40){\ing{sunem10q10ma6s}} %\put(24,20){\ing{nem120ma}} \put(30,20){\ing{sunem0h}} \put(12,514){\sx{3.1}{$y$}} \put(12,417){\sx{3}{$4$}} \put(12,317){\sx{3}{$3$}} \put(12,217){\sx{3}{$2$}} \put(12,117){\sx{3}{$1$}} \put(12,017){\sx{3}{$0$}} %\put(-6,417){\sx{3}{$-1$}} %\put(-6,317){\sx{3}{$-2$}} %\put(-6,217){\sx{3}{$-3$}} %\put(-6,117){\sx{3}{$-4$}} %\put(-6, 17){\sx{3}{$-5$}} \put( 9, 0){\sx{3}{$-5$}} \put(109, 0){\sx{3}{$-4$}} \put(209, 0){\sx{3}{$-3$}} \put(309, 0){\sx{3}{$-2$}} \put(409, 0){\sx{3}{$-1$}} \put(530, 0){\sx{3}{$0$}} \put(630, 0){\sx{3}{$1$}} \put(730, 0){\sx{3}{$2$}} \put(830, 0){\sx{3}{$3$}} \put(930, 0){\sx{3}{$4$}} \put(1022,0){\sx{3.2}{$x$}} % \put(788,384){\sx{3}{\rot{57} $u\!=\!0.1$\ero}} \put(620,498){\sx{3}{\rot{-73} $u\!=\!0.2$\ero}} \put(150,429){\sx{3}{\rot{16} $u\!=\!0.3$\ero}} \put(166, 32){\sx{3}{\rot{79} $u\!=\!0.4$\ero}} \put(308, 36){\sx{3}{\rot{74} $u\!=\!0.5$\ero}} \put(392, 35){\sx{3}{\rot{70} $u\!=\!0.6$\ero}} \put(445, 31){\sx{3}{\rot{66} $u\!=\!0.7$\ero}} \put(484, 29){\sx{3}{\rot{64} $u\!=\!0.8$\ero}} \put(538, 26){\sx{3}{\rot{68} $u\!=\!1$\ero}} % %\put(642,76){\sx{3}{\rot{72} $u\!=\!0.2$\ero}} %\put(784,148){\sx{3}{\rot{-59} $u\!=\!0.1$\ero}} % \put(664,376){\sx{3}{\rot{-9} $v\!=\!0.4$\ero}} \put(520,446){\sx{3}{\rot{39} $v\!=\!0.3$\ero}} \put(308,416){\sx{3}{\rot{-84} $v\!=\!0.2$\ero}} \put(110,282){\sx{3}{\rot{-41} $v\!=\!0.1$\ero}} \put(60, 18){\sx{3}{\rot{0} $v\!=\!0$\ero}} \put(858, 19){\sx{3}{\rot{0} \bf huge values\ero}} %\put(100,236){\sx{3}{\rot{42} $v\!=\!-0.1$\ero}} %\put(321, 110){\sx{3}{\rot{82} $v\!=\!-0.2$\ero}} % \end{picture} \end{document}