File:Nembraplot.jpg

Parametric plot of position of the branch point of the ArcNem function:

$x\!+\!\mathrm i y = \mathrm{NemBra}(q)$

for positive $q$. Function NemBra is solution $z=\mathrm{NemBra}(q)$ of equation

$\mathrm{Nem}_q'(z)=0$

where the Nemtsov function Nem is special polynomial

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

and, correspondently,

$\mathrm{Nem}_q'(z)=1+3z^2+4qz^3$

Function NemBra is necessary for implementation of function ArqNem, that, in its turn, happen to be necessary for implementation of the non-integer iterates of the Nemtsov function, suggested as example of exotic iterates for the book Superfunctions.

Mathematica generator of the solution
T[z_] = z + z^3 + q z^4 s123 = Solve[T'[z] == 0, z] s1[q_] = ReplaceAll[z, Extract[s123, 1]] Simplify[T'[ 1/4 (-(1/q) + 1/( q (-1 - 8 q^2 - 4 Sqrt[q^2 + 4 q^4])^(1/3)) + (-1 - 8 q^2 - 4 Sqrt[q^2 + 4 q^4])^(1/3)/q)]] Print[N[Normal[Simplify[Series[s1[q], {q, 0, 7}]]], 16]] Simplify[Series[s1[q], {q, Infinity, 4}]]

C++ generator of curve
typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)

void ado(FILE *O, int X, int Y) {      fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%'); fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y); fprintf(O,"/M {moveto} bind def\n"); fprintf(O,"/L {lineto} bind def\n"); fprintf(O,"/S {stroke} bind def\n"); fprintf(O,"/s {show newpath} bind def\n"); fprintf(O,"/C {closepath} bind def\n"); fprintf(O,"/F {fill} bind def\n"); fprintf(O,"/o {.002 0 360 arc C S} bind def\n"); fprintf(O,"/times-Roman findfont 20 scalefont setfont\n"); fprintf(O,"/W {setlinewidth} bind def\n"); fprintf(O,"/RGB {setrgbcolor} bind def\n");} //#include "ado.cin"

z_type nem(DB q,z_type z){ return z*(1.+z*z*(1.+z*q)); } z_type nem1(DB q,z_type z){ return 1.+z*z*(3.+z*(4.*q)); } z_type nembra0(DB q){ return -0.5773502691896258*I+ q*(2./9+       q*(0.2138334330331947*I+ q*(-0.2633744855967078 +       q*(-0.3658927631901332*I+ q*(0.5462581923487273 +       q*(0.8556857213229570*I+ q*(-1.387322393266609       ))))))) ;}

z_type nembrao(DB q){ z_type x,y,z,s; x=conj(pow(z_type(-.25/q,0.),1./3.)); y=x*x; z=y*y; s=1.+y*(1.+y*(1.+y*(2./3.+z*(-2./3.+y*(-7./9.+z*(11./9.+y*(130./91.) )    )     )        )         )         ); return s*x;}

z_type nembra(DB q){ if(fabs(q)<.021) return nembra0(q); if(fabs(q) >20.) return nembrao(q); z_type Q,v,V; Q=q*q; v=-1.-8.*Q+4.*sqrt(Q+4.*Q*Q); V=pow(v,1./3.); return (.25/q)*(-1.+1./V+V); }

int main{ int m,n; z_type z,c,d,e; DB x,y,q,u,v;

DO(n,41){ q=-.04+.01*n; c=nembra0(q); d=nem(q,c); e=nem1(q,c); printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );} printf("\n"); DO(n,41){ q=.1*n; c=nembra(q); d=nem(q,c); e=nem1(q,c); printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );} printf("\n"); DO(n,41){ q=1.+1.*n; c=nembrao(q); d=nem(q,c); e=nem1(q,c); printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}

//FILE *o; o=fopen("08.eps","w"); ado(o,154,604); FILE *o; o=fopen("nembraplo.eps","w"); ado(o,154,604); fprintf(o,"2 2 translate 1000 1000 scale 2 setlinecap\n");
 * 1) define M(x,y) fprintf(o,"%8.6f %8.6f M\n",(0.+x),(0.+y));
 * 2) define L(x,y) fprintf(o,"%8.6f %8.6f L\n",(0.+x),(0.+y));
 * 3) define o(x,y) fprintf(o,"%8.6f %8.6f o\n",(0.+x),(0.+y));

for(n=0;n<61;n+=5){ M(0,.01*n) L(.15,.01*n) } for(n=0;n<31;n+=5){ M(.01*n,0) L(.01*n,.6) } fprintf(o,".0006 W S 1 setlinejoin\n"); fprintf(o,"1 0 0 RGB .001 W\n");

for(n= 1;n<10;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } for(n=11;n<20;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } for(n=21;n<30;n++){ q=.1*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } fprintf(o,"0 .5 0 RGB\n"); for(n= 1;n<20;n++){ q=1.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } fprintf(o,"0 0 1 RGB\n"); for(n= 1;n<10;n++){ q=10.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } fprintf(o,".5 0 .8 RGB\n"); for(n= 1;n<10;n++){ q=100.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } fprintf(o,"0 0 0 RGB\n"); for(n= 0;n<11;n+=10){ q=100.*n; c=nembra(q); x=Re(c); y=-Im(c); o(x,y); } q=10000.; c=nembra(q); x=Re(c); y=-Im(c); o(x,y);

for(n= 0;n<400;n++){ q=-.006+.04*(n+.01*n*n+.001*n*n*n); c=nembra(q); x=Re(c); y=-Im(c); if(n==0) M(x,y) else L(x,y); printf("%9.3lf %16.6lf %16.6lf\n",q,x,y); } L(0,0) fprintf(o,"0 0 0 RGB .001 W 0 setlinecap S\n");

fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o);

system("epstopdf nembraplo.eps"); system("open    nembraplo.pdf"); getchar; return 0;}

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 174pt \paperheight 620pt \topmargin -100pt \oddsidemargin -72pt \textheight 800px \parindent 0pt \usepackage{graphicx} \usepackage{rotating} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \begin{document} \begin{picture}(170,610) \put(20,14){\ing{nembraplo}} \put(8,612){\sx{1.4}{$y$}} \put(0,512){\sx{1.4}{$0.5$}} \put(0,412){\sx{1.4}{$0.4$}} \put(0,312){\sx{1.4}{$0.3$}} \put(0,212){\sx{1.4}{$0.2$}} \put(0,112){\sx{1.4}{$0.1$}} \put(8,12){\sx{1.4}{$0$}} \put(20,1){\sx{1.4}{$0$}} \put(60,1){\sx{1.4}{$0.05$}} \put(112,1){\sx{1.4}{$0.1$}} \put(165,2){\sx{1.4}{$x$}} %\put(18,601){\sx{1.1}{$q\!=\!0$}} \put(0,583){\sx{1.1}{$q\!=\!0$}} %\put(63,593){\sx{1.1}{$q\!=\!0.2$}} \put(30,576){\sx{1.1}{$q\!=\!0.2$}} %\put(97,574){\sx{1.1}{$q\!=\!0.4$}} \put(62,558){\sx{1.1}{$q\!=\!0.4$}} %\put(122,550){\sx{1.1}{$q\!=\!0.6$}} \put(85,535){\sx{1.1}{$q\!=\!0.6$}} %\put(137,528){\sx{1.1}{$q\!=\!0.8$}} \put(100,514){\sx{1.1}{$q\!=\!0.8$}} %\put(148,506){\sx{1.1}{$q\!=\!1$}} \put(121,494){\sx{1.1}{$q\!=\!1$}} \put(137,424){\sx{1.1}{$q\!=\!2$}} \put(138,380){\sx{1.1}{$q\!=\!3$}} \put(136,350){\sx{1.1}{$q\!=\!4$}} \put(133,330){\sx{1.1}{$q\!=\!5$}} \put(129,310){\sx{1.1}{$q\!=\!6$}} \put(122,286){\sx{1.1}{$q\!=\!8$}} \put(110,268){\sx{1.1}{$q\!=\!10$}} \put( 90,214){\sx{1.1}{$q\!=\!20$}} \put( 82,192){\sx{1.1}{$q\!=\!30$}} \put( 88,127){\sx{1.1}{$q\!=\!100$}} \put( 77,104){\sx{1.1}{$q\!=\!200$}} \put( 54,63){\sx{1.1}{$q\!=\!1000$}} \put( 39,36){\sx{1.1}{$q\!=\!10000$}} \end{picture} \end{document}