File:Sinplo2t100.jpg

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.*/

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

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

FILE *o;o=fopen("sinplo2.eps","w"); ado(o,318,160);
 * 1) define M(x,y) {fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);}
 * 2) 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}