File:Straroplot.jpg

Esplicit plot of function StraRo for various vvalues of parameter:

$q\!=\!0.5$, red

$q\!=\!1$, green

$q\!=\!2$, blue

Dependences $y=\mathrm{StraRo}_q(x)$ are shown with thick lines. These lines have jump at zero. Such a jump is unavoidable wile function StraRo is holomorphic above the upper branch point and below the lower branch point. However, the jump can be moved away from the real axis in the complex plane; the corresponding extensions are shown with thin lines.

C++ generator of curves
//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); }

z_type F1(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 ; return sqrt(r); }

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; FILE *o;o=fopen("straroplo.eps","w"); ado(o,604,724); //FILE *o;o=fopen("05.eps","w"); ado(o,604,724); fprintf(o,"302 122 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(m=-30;m<31;m+=10){M(.1*m,-1.2) L(.1*m,6) } fprintf(o,".003 W 0 0 0 RGB S\n");

for(n=-12;n<61;n+=1){M(-3,.1*n) L(3,.1*n)} fprintf(o,".0006 W 0 0 0 RGB S\n");

for(n=-10;n<61;n+=10){M(-3,.1*n) L(3,.1*n)} fprintf(o,".01 W 0 0 0 RGB S 0 setlinecap\n");

Q=0.5; S=nembran(Q); DO(m,301){x=.01*(m+.1); y=.1*Re(F(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".02 W 1 0 0 RGB S\n");

DO(m,301){x=-.01*(m); y=.1*Re(F(x+.001*I)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; //printf("%8.2lf %8.2lf\n",x,y); } fprintf(o,".02 W 1 0 0 RGB S\n");

DO(m,301){x=-3.+.01*(m+.1); y=.1*Re(F1(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".001 W 0 0 0 RGB S\n");

Q=1.; S=nembran(Q); DO(m,301){x=.01*(m+.1); y=.1*Re(F(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".02 W 0 .8 0 RGB S\n"); DO(m,301){x=-.01*(m); y=.1*Re(F(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".02 W 0 .8 0 RGB S\n"); DO(m,301){x=-3.+.01*(m+.1); y=.1*Re(F1(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".001 W 0 0 0 RGB S\n");

Q=2.; S=nembran(Q); DO(m,301){x=.01*(m+.1); y=.1*Re(F(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".02 W 0 0 1 RGB S\n"); DO(m,301){x=-.01*(m); y=.1*Re(F(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".02 W 0 0 1 RGB S\n"); DO(m,301){x=-3.+.01*(m+.1); y=.1*Re(F1(x)); if(m==0) M(x,y) else L(x,y) ; if(y>8) break; } fprintf(o,".001 W 0 0 0 RGB S\n");

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

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

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 606pt \paperheight 726pt \topmargin -102pt \oddsidemargin -72pt \textheight 800px \parindent 0pt \usepackage{graphicx} \usepackage{rotating} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \begin{document} \begin{picture}(170,718) %\put(20,14){\ing{nembraplo}} \put(277,706){\sx{3.3}{$y$}} \put(260,612){\sx{3.3}{$50$}} \put(260,511){\sx{3.3}{$40$}} \put(260,411){\sx{3.3}{$30$}} \put(260,311){\sx{3.3}{$20$}} \put(260,211){\sx{3.3}{$10$}} \put(274,111){\sx{3.3}{$0$}} \put(224, 11){\sx{3.3}{$-10$}} \put( 74,94){\sx{3.3}{$-2$}} \put(174,94){\sx{3.3}{$-1$}} \put(294,94){\sx{3.3}{$0$}} \put(394,94){\sx{3.3}{$1$}} \put(494,94){\sx{3.3}{$2$}} \put(584,95){\sx{3.33}{$x$}} \put(0,0){\ing{straroplo}} \end{picture} \end{document}