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Complex map of function StraRo:

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

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


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 $q\!=\!2$.

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 $q=2$, the evaluation suggests

$x_0=0.089910637427530 ~$, $~ y_0= 0.296978217844034$


C++ generator of map

Files ado.cin, conto.cin, nembran.cin should be loaded in order to compile the C++ code below:

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
//using namespace std;
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"

DB Q=2;
z_type T(z_type z){ return z*(1.+z*z*(1.+z*Q));}

#include "nembran.cin"

z_type S=nembran(Q);

z_type F(z_type z){ z_type a,b,r,R ; a=Q-z; b=1. + 4.*Q*z; r = 27.*a*a + 4.*b*b*b ;
DB x,y; x=Re(z); y=Im(z);
if( x<=0 ) return -sqrt(r);
if( y > 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("straro20a.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");

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;
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);


// 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");

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"

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

fclose(o); free(f); free(g); free(w);

      system("epstopdf straro20a.eps");
      system( "open straro20a.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh

Latex generator of labels

\paperwidth 4200pt
\paperheight 4200pt
\topmargin -100pt
\oddsidemargin -80pt
\textheight 4800px
\parindent 0pt
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\put(-30, 970){\sx{14}{$-1$}}
\put( 50,-120){\sx{14}{$-2$}}
\put(330,2000){\sx{14}{\bf cut}}
\put(2230,2060){\rot{69}\sx{14}{\bf cut}\ero}
\put(800, 3160){\rot{-72}\sx{14}{$v\!=\!-40$}\ero}%
\put(1140, 3300){\rot{-64}\sx{14}{$v\!=\!-10$}\ero}%
\put(1260, 3360){\rot{-64}\sx{14}{$v\!=\!0$}\ero}%
\put(1360, 3420){\rot{-60}\sx{14}{$v\!=\!10$}\ero}%
\put(400, 1870){\rot{-5}\sx{14}{$u\!=\!-10$}\ero}
\put(400, 1520){\rot{-7}\sx{14}{$u\!=\!-40$}\ero}
\put(760, 720){\rot{66}\sx{14}{$v\!=\!40$}\ero}
\put(1090, 600){\rot{65}\sx{14}{$v\!=\!10$}\ero}
\put(1200, 500){\rot{58}\sx{14}{$v\!=\!0$}\ero}
\put(1320, 370){\rot{57}\sx{14}{$v\!=\!-10$}\ero}
\put(3300, 1440){\rot{10}\sx{14}{$v\!=\!-40$}\ero}

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