File:Straro05at.jpg

Complex map of function StraRo for parameter $q\!=\!0.5$

$u\!+\!\mathrm i v=\mathrm{Straro}_q(x\!+\!\mathrm i y)$

For positive $q$ and complez $z$, function $\mathrm{Straro}_q(z)$ is such that

$\mathrm{StraRo}_q(z)^2=27(z\!-\!q)^2+4(1\!+\!4qz)^3$

where $q\!>\!0$ and the cut lines are chosen in the special way:

From the negative branch point, there is cut line along the negative part of the real axis to $-\infty$.

Also, from each complex branch point, there is straight line to zero.

In the rest of the complex plane, for positive $q$, function $\mathrm{StraRo}$ is holomorphic.

The figure shows the map for $q\!=\!1$.

The special choice of the cuts above is used to define function ArcNem (with the same structure of cut lines), and it is used for evaluation of function AuNem, which is Abel function for the Nemtsov function. This function is proposed al illustration of the exotic iterates of the Nemtsov function constructed as zero.

The upper complex branch point is expressed with function NemBran,

$x_0\!+\!\mathrm i y_0=\mathrm{NemBran}(q)$

The complex brach points are shown with additional grid lines $x=x_0$ and $y=\pm y_0$

For $q=1$, the evaluation suggests

$x_0 \approx 0.04746498715101~$, $~ y_0 \approx 0.368104471973035$

C++ generator of map
Files ado.cin, conto.cin, nembran.cin should be loaded in order to compile the C++ 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 Im(S)     )        return I*sqrt(-r); if( y     < - Im(S)   )        return -I*sqrt(-r); if( x*Im(S)< fabs(y)*Re(S) )   return -sqrt(r); return sqrt(r); }

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; 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 FILE *o;o=fopen("straro05a.eps","w"); ado(o,4002,4002); fprintf(o,"2001 2001 translate\n 1000 1000 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-2+.004*(m-.5); DO(n,N1) Y[n]=-2+.008*(n-.5); //for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); } for(m=-20;m<21;m+=5){M(.1*m,-2) L(.1*m,2) } for(n=-20;n<21;n+=5){M( -2,.1*n) L(2,.1*n)} fprintf(o,".001 W 0 0 0 RGB S\n");

//z=nembran(Q); x=Re(S); y=Im(S); M(x,-1)L(x,1) M(-1,y)L(1,y) M(-1,-y)L(1,-y) fprintf(o,".004 W .5 .5 0 RGB S\n"); printf("x,y %18.15lf %18.15lf\n",x,y);

DO(m,M1)DO(n,N1){ g[m*N1+n]=99999999; f[m*N1+n]=99999999;} DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)

// c=-1./z; // c=-(1.+3.*z*z)/(4.*z*z*z);

//c=Fru(z); c=F(z);

// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p; p=Re(c); q=Im(c); if(p>-100000 && p<100000 && q>-100000 && q<100000 ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=4;q=.5;

for(m=-10;m<10;m++)for(n=2;n<10;n+=2)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<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".001 W .8 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".001 W 0 0 .8 RGB S\n"); for(m= 1;m<41;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<41;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,".003 W .6 0 .6 RGB S\n"); for(m=-40;m<41;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".004 W 0 0 0 RGB S\n"); // fprintf(o,"0 setlinejoin 0 setlinecap\n"); // M(-10,0)L(0,0) fprintf(o,"1 1 1 RGB .02 W S\n"); //#include "plofu.cin"

x=Re(S); y=Im(S); M(x,y)L(0,0) L(x,-y) fprintf(o,".01 W 1 1 .5 RGB 0 setlinecap S\n"); printf("x,y %18.15lf %18.15lf\n",x,y);

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w);

system("epstopdf straro05a.eps"); system(   "open straro05a.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh } //

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 4200pt \paperheight 4200pt \topmargin -100pt \oddsidemargin -80pt \textheight 4800px \parindent 0pt \usepackage{graphicx} \usepackage{rotating} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \begin{document} \begin{picture}(4090,4050) %\put(20,14){\ing{nembraplo}} %\put(20,14){\ing{nembran}} %\put(190,30){\ing{straro10a}} \put(190,30){\ing{straro05a}} \put(70,3940){\sx{14}{$y$}} \put(70,2970){\sx{14}{$1$}} \put(70,1970){\sx{14}{$0$}} \put(-30, 970){\sx{14}{$-1$}} \put(-30,-20){\sx{14}{$-2$}} \put( 50,-120){\sx{14}{$-2$}} \put(1050,-120){\sx{14}{$-1$}} \put(2170,-120){\sx{14}{$0$}} \put(3170,-120){\sx{14}{$1$}} \put(4110,-120){\sx{14}{$x$}}

\put(820,3330){\rot{40}\sx{14}{$u\!=\!-14$}\ero}% \put(1610,3550){\rot{55}\sx{14}{$u\!=\!-10$}\ero}% \put(2540,3300){\rot{68}\sx{14}{$u\!=\!0$}\ero}% \put(3310,2960){\rot{74}\sx{14}{$u\!=\!10$}\ero}

\put(1600,1760){\rot{90}\sx{14}{$u\!=\!-6$}\ero} \put(1790,1760){\rot{90}\sx{14}{$u\!=\!-5$}\ero}

\put(240,2000){\sx{14}{$v\!=\!0$}} \put(710,2000){\sx{14}{$v\!=\!0$}} \put(3330,2000){\sx{14}{$v\!=\!0$}} \put(2250,2060){\rot{80}\sx{14}{\bf cut}\ero} \put(540, 2480){\rot{-86}\sx{14}{$v\!=\!0$}\ero} \put(620, 1590){\rot{84}\sx{14}{$v\!=\!0$}\ero} % \put(940, 2360){\rot{-20}\sx{14}{$v\!=\!1$}\ero} \put(940, 1630){\rot{20}\sx{14}{$v\!=\!-1$}\ero} \put(1100, 1450){\rot{20}\sx{14}{$v\!=\!-2$}\ero} \put(1210, 1300){\rot{20}\sx{14}{$v\!=\!-3$}\ero} \put(1300, 1190){\rot{20}\sx{14}{$v\!=\!-4$}\ero} % \put(3390,2820){\rot{-11}\sx{14}{$v\!=\!10$}\ero} \put(3400,1180){\rot{8}\sx{14}{$v\!=\!-10$}\ero} \end{picture} \end{document}