File:Nembraplot.jpg

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}]]

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#define DB double
#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");
#define M(x,y) fprintf(o,"%8.6f %8.6f M\n",(0.+x),(0.+y));
#define L(x,y) fprintf(o,"%8.6f %8.6f L\n",(0.+x),(0.+y));
#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;}

\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}