File:Susinplot.jpg

Summary

Explicit plot of function SuSin is shown with the thick curve, \(y\!=\! \mathrm{SuSin}(x)\).

For comparison, the thin curve shows the asymptotic of SuSin, \(y \!=\! \sqrt{3/x}\).

SuSin is Superfunction of sinus.

This plot is used as Fig.12.1 at page 148 of book «Superfunctions», 2020 ^[1][2]
as a support of the pretentious claim that the Author can construct a real-holomorphic superfunction for any growing real-holomorphic transfer function, and also the Abelfunction, and with these two functions can iterate the transfer function \(n\) times, and the number \(n\) of the iterate has no need to be integer.
The similar claim refers to the colleagues who had read the book cited; they also can do the same using the tools described in the Book.

So, look at the book for the description.

The numeric implementation of SuSin below is not yet a final tool, because it provides only 6 significant figures. In addition, the complex double implementation of ArcSin used is not robust. In such a way, the algorithm below allows to plot the camera–ready figures, but it may be not sufficient for some other applications. However I want to keep the generator simple, following the last 6th of the TORI axioms - until more digits are requested for some application.

C++ generator of curves

// Files ado.cin, arcsin.cin, and susin.cin should be loaded to the 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;
#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"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
DB x0=0.;
DO(m,14){y=Re(susin(z_type(1.,1.e-9)+x0))-1.;
        x0+=4.*y;
        printf("%2d %19.16f %19.16f\n",m,x0,y);}

FILE *o;o=fopen("susinplot1.eps","w"); ado(o,1002,242);
#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<11;m++){M(m,-1) L(m,2) }
for(n=0;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");

M(0,M_PI/2.);
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,".03 W 0 0 .8 RGB S\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,".01 W 0 0 0 RGB S\n");

//n=0;DO(m,100){ x=.1*m; z=z_type(x,1.e-8); c=susin(z); y=Im(c); if(y>-2 && y<3) { if(n==0) M(x,y) else L(x,y); n++;}}
//fprintf(o,".02 W .8 0 0 RGB S\n"); printf("n=%3d\n",n);

fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);
      system("epstopdf susinplot1.eps"); 
      system( "open susinplot1.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\paperwidth 1026pt 
\paperheight 225pt
\topmargin -109pt
\oddsidemargin -90pt
\newcommand \sx {\scalebox}
\pagestyle{empty}
\begin{document}
\begin{picture}(1016,204)
\put(20,1){\includegraphics{susinplot1}}
\put(2,191){\sx{2.4}{$y$}}
\put(-1,151){\sx{2.8}{$\frac{\pi}{2}$}}
\put(2,93){\sx{2.4}{$1$}}
\put(2,-5){\sx{2.4}{$0$}}
\put(15,-19){\sx{2.4}{$0$}}
\put(115,-19){\sx{2.4}{$1$}}
\put(215,-19){\sx{2.4}{$2$}}
\put(315,-19){\sx{2.4}{$3$}}
\put(415,-19){\sx{2.4}{$4$}}
\put(516,-19){\sx{2.4}{$5$}}
\put(616,-19){\sx{2.4}{$6$}}
\put(717,-19){\sx{2.4}{$7$}}
\put(817,-19){\sx{2.4}{$8$}}
\put(917,-19){\sx{2.4}{$9$}}
\put(1010,-19){\sx{2.5}{$x$}}
%\put(45,134){\sx{2.5}{$y\!=\!\mathrm{SuSin}(x)$}}
\put(140,166){\sx{2.8}{$y\!=\! \sqrt{3/x}$}}
\put(135,56){\sx{2.8}{$y\!=\!\mathrm{SuSin}(x)$}}
\end{picture}
\end{document}

References

Keywords

«AuSin», «Exotic iteration», «Fixed point», «Iterate», «Iteration», «Sin», «Superfunction», «Superfunctions», «SuSin», «Transfer function», «Transferfunction», «[[]]»,

«Суперфункции»,