Difference between revisions of "File:Sinplo2t100.jpg"

Fig.12.6 at from page 157 of book «Superfunctions» ^[1]

Iterates of function sin of real argument, explicit plot

\(y=\sin^n(x)=\mathrm{SuSin}\big(n+\mathrm{AuSin}(x)\big)\)

for various real values of number \(n\) of iterate.

For the evaluation at non-integer \(n\), the representation through the superfunction SuSin and the Abel function AuSin are used.

Example:

\(\sin^{1/2}(\pi/2)\approx 1.140179476170028\)

\(\sin^{1/2}(1.140179476170028) \approx 1.000000000000003\approx 1\)

This figure is almost the same as http://mizugadro.mydns.jp/t/index.php/File:Sinplo1t100.jpg , but the thin curves are made a little bit thicker, to make them seen at the poor resolution of the screen.

C++ generator of curves

/* Files ado.cin, arcsin.cin, susin.cin, ausin.cin should be loaded in order to compile the 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;
#include<complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

#include "ado.cin"
#include "arcsin.cin"
#include "susin.cin"
#include "ausin.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;

FILE *o;o=fopen("sinplo2.eps","w"); ado(o,318,160);
#define M(x,y) {fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);}
#define L(x,y) {fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);}

fprintf(o,"1 1 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(m=0;m<4;m++){M(m,0) L(m,M_PI/2.) }
for(n=0;n<2;n++){M( 0,n) L(M_PI,n)}
fprintf(o,".004 W 0 0 0 RGB S\n");
M(M_PI/2.,0); L(M_PI/2.,M_PI/2)
M(M_PI,0); L(M_PI,M_PI/2)
M(0,M_PI/2.); L(M_PI,M_PI/2)
fprintf(o,".001 W 0 0 0 RGB S\n");
M(0,0) L(M_PI/2.,M_PI/2.)
fprintf(o,".007 W 0 1 1 RGB S\n");
M(0,0) L(M_PI/2., M_PI/2.) L(M_PI,0)
fprintf(o,".001 W 0 0 0 RGB S\n");

fprintf(o,"1 setlinejoin 1 setlinecap\n");


M(0,0) DO(m,158){x=.01*(m+.1);y=sin(x);                 L(y,x); } fprintf(o,".009 W 0 1 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(sin(x));            L(y,x); } fprintf(o,".009 W 0 1 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(sin(sin(x)));       L(y,x); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(sin(sin(sin(x))));  L(y,x); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(sin(sin(sin(sin(x)))));L(y,x); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(x); DO(n, 10)y=sin(y); L(y,x); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(x); DO(n, 20)y=sin(y); L(y,x); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,158){x=.01*(m+.1);y=sin(x); DO(n,100)y=sin(y); L(y,x); } fprintf(o,".006 W 0 0 1 RGB S\n");

M(0,0) DO(m,315){x=.01*(m+.1);y=sin(x);                 L(x,y); } fprintf(o,".009 W 0 1 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(sin(x));            L(x,y); } fprintf(o,".009 W 0 1 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(sin(sin(x)));       L(x,y); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(sin(sin(sin(x))));  L(x,y); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(sin(sin(sin(sin(x)))));L(x,y); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(x); DO(n, 10)y=sin(y); L(x,y); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(x); DO(n, 20)y=sin(y); L(x,y); } fprintf(o,".006 W 0 0 1 RGB S\n");
M(0,0) DO(m,315){x=.01*(m+.1);y=sin(x); DO(n,100)y=sin(y); L(x,y); } fprintf(o,".006 W 0 0 1 RGB S\n");

for(n=1;n<21;n++){
        M(0,0) DO(m,158){ x=.01*(m+.1); z=x; c=ausin(z); c=susin(.1*n+c); y=Re(c); 
        if(abs(Im(c))>1.e-9) break; if(y>-1. && y<4.) L(y,x) else break; } 
        fprintf(o,".003 W 0 0 0 RGB S\n");
        }

for(n=1;n<21;n++){
        M(0,0) DO(m,315){ x=.01*(m+.1); z=x; c=ausin(z); c=susin(.1*n+c); y=Re(c); 
        if(abs(Im(c))>1.e-9) break; if(y>-1. && y<4.) L(x,y) else break; } 
        fprintf(o,".003 W 0 0 0 RGB S\n");
        }

fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);
      system("epstopdf sinplo2.eps"); 
      system(    "open sinplo2.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 3230pt 
\paperheight 1700pt
\topmargin -100pt
\oddsidemargin -72pt
\textwidth 3200pt
\textheight 1700pt
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\pagestyle{empty}
\begin{document}
\sx{10}{\begin{picture}(328,168)
\put(4,9){\includegraphics{sinplo2}}
\put(-1.6,162){\sx{1.}{$y$}}
\put(-1.6,106){\sx{1.}{$1$}}
\put(-1.6, 06){\sx{1.}{$0$}}
%\put(-7, 06){\sx{1.2}{$-2$}}
\put(3,0){\sx{1.}{$0$}}
\put(103,0){\sx{1.}{$1$}}
\put(153,0){\sx{1.}{$\pi/2$}}
\put(203,0){\sx{1.}{$2$}}
\put(303,0){\sx{1.}{$3$}}
\put(315,0){\sx{1.}{$x$}}
\put(24.6,115){\sx{.8}{\rot{89}$n\!=\!-100$\ero}}
\put(42.6,115){\sx{.8}{\rot{89}$n\!=\!-20$\ero}}
\put(53,115){\sx{.8}{\rot{87}$n\!=\!-10$\ero}}
\put(67,115){\sx{.8}{\rot{85}$n\!= -5$\ero}}
%\put(70,115){\sx{.8}{\rot{82}$n\!=\!-4$\ero}}
\put(77,115){\sx{.8}{\rot{82}$n\!=\!-3$\ero}}
\put(84.3,115){\sx{.8}{\rot{79}$n\!=\!-2$\ero}}
\put(94,112){\sx{.8}{\rot{68}$n\!=\!-1$\ero}}
\put(123.4,134){\sx{.8}{\rot{63}$n\!=\! -0.2$\ero}}
\put(146,147){\sx{.9}{\rot{45}$n\!=\!0$\ero}}
\put(142,140){\sx{.8}{\rot{10}$n\!= 0.1$\ero}}
\put(140,131.6){\sx{.8}{\rot{10}$n\!= 0.2$\ero}}
\put(139,122){\sx{.8}{\rot{9}$n\!= 0.4$\ero}}
\put(110,95){\sx{.8}{\rot{21}$n\!= 1$\ero}}
\put(110,84.3){\sx{.8}{\rot{13}$n\!= 2$\ero}}
\put(110,77.3){\sx{.8}{\rot{8}$n\!= 3$\ero}}
%\put(110,72){\sx{.8}{\rot{6}$n\!= 4$\ero}}
\put(110,67){\sx{.8}{\rot{6}$n\!= 5$\ero}}
\put(110,53){\sx{.8}{$n\!=\! 10$}}
\put(110,42.3){\sx{.8}{$n\!= 20$}}
\put(110,24.3){\sx{.8}{$n\!=\!100$}}
\end{picture}}
\end{document}

Keywords

«Abel function», «Abelfunction», «Iteration», «Sin», «SuSin», «Superfunction», «Superfunctions»,

«Суперфункции»,

References