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Explicit plot of function SuSin in comparison to its approximations.

Black think solid curve shows $y= \mathrm{SuSin}(x)$, this is superfunction of sin, id est, super sin.

The upper thin blue curves shows the leading term of its asymptotic expansion, $y=\sqrt{3/x}$, suggested in 2012 by Kursernas Hemsidor [1].

The red dashed curve shows the approximation suggested in 2012 by Thomas Curtright [2]

$y=\exp\Big( \big(1\!-\!\sqrt{x}\big)\ln(\pi/2)\Big)$

The lowest red thin curve shows the difference between the $SuSin(x)$ and the approximation by Thomas, scaled with factor 10.


  1. Kursernas Hemsidor. 273027 Introduction to Dynamical Systems 2012. Derivation of Niklas Carlsson;s formula. Cited by the state for December 2013: Let function f(x) be sin(x). We want to evaluate, approximately, the value of the nth iterate of f(x) ... If the formula is correct, it will take 3⋅1010 .. steps to reat 0.00001 from 1. .. $f^n(x)\approx \sqrt{\frac{3}{n}}$, $x\approx 1$, $n$ lage ... How good is this formula? after 60000 iterations of the value 0.0071 and the discrepancy -4.7s-007 ..
  2. Dr. Thomas Curtright. Continuous iterates continue to be interesting, after 150 years of study. (2012) As a first illustration, we display the continuous iterates of the sine function, sin[t](x). Note that the maximum values at x = π/2 are approximately given by exp[(1-√t) ln(π/2)].

C++ generator of curves

//Files ado.cin, arcsin.cin, and susin.cin should be loaded to working directory 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;
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"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
DB x0=0.;
        printf("%2d %19.16f %19.16f\n",m,x0,y);}
//FILE *o;o=fopen("susinplot1.eps","w"); ado(o,1002,244);
//FILE *o;o=fopen("04.eps","w"); ado(o,1002,348);
FILE *o;o=fopen("susinploa.eps","w"); ado(o,1002,348);
#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,"1 106 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(m=0;m<11;m++){M(m,-1) L(m,2) }
for(n=-1;n<3;n++){M( 0,n) L(10,n)}
fprintf(o,".006 W 0 0 0 RGB S\n");
M(0,M_PI/2.); L(10,M_PI/2)
fprintf(o,".004 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 1 setlinecap\n");
DO(m,100){ x=.5+.1*m; y=sqrt(3./x); if(m==0) M(x,y) else L(x,y) ; if ( x>10.) break;}
fprintf(o,".006 W 0 0 1 RGB S\n");
fprintf(o,"1 setlinejoin 0 setlinecap\n");
DO(m,300){ x=.0001+.04*m/(1+5./(.3+m)); y=exp((1.-sqrt(x))*log(M_PI/2)); if(m/2*2==m) M(x,y) else L(x,y) ; if ( x>10.) break;}
fprintf(o,".02 W 1 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,2002){ x=.005*(m+.3); z=z_type(x,1.e-8); c=susin(z); y=Re(c); L(x,y); printf("%8.5f %8.5f\n",x,y); }
fprintf(o,".012 W 0 0 0 RGB S\n");
DO(m,20022){ x=.005*(m+.3); z=z_type(x,1.e-8); c=susin(z);
            y=exp((1.-sqrt(x))*log(M_PI/2))- Re(c); y*=10; L(x,y); printf("%8.5f %8.5f\n",x,y);
if(x>10) break; }
fprintf(o,".006 W 1 0 0 RGB S\n");

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

Latex generator of labels

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\paperwidth 1026pt
\paperheight 345pt
\topmargin -109pt
\oddsidemargin -90pt
\newcommand \sx {\scalebox}
\put(190,246){\sx{1.8}{\rot{-12}$y\!=\! \sqrt{3/x}$\ero}}

\put(190,110){\sx{1.8}{\rot{-8}$y\!=\! 10\Big(\exp((1\!-\!\sqrt{x})\ln(\pi/2))-\mathrm{SuSin}(x)\Big)$\ero}}

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