Difference between revisions of "File:Shelr80.png"

Central picture of Fig.18.3 at page 250 of book «Superfunctions» ^[1], 2020.

The picture shows the explicit plot of tetration to Sheldon base for real values of the argument.

The Sheldon base is defined with \[ b= 1.52598338517+0.0178411853321 \,\mathrm i \]

Sheldon Levenstein had suggested this number, but he did not provide any way of evaluation of this number; so this value is considered as exact. Sheldon Levenstein asked to plot pictures for the Tetration to this base. No other similar request had been received; so, the request by Sheldon Levenstein had been proceeded. The tetration had been constricted with the Cauchi intetral, in the similar way as it is done for the natural tetration ^[2].

The blue curve shows \(y=\Re\Big( \mathrm{tet}_b(x)\Big)\)

The red curve shows \(y=\Im\Big( \mathrm{tet}_b(x)\Big)\)

For comparison, the black line shows the graphic for real base \(b=1.5\), id est, \(y=\mathrm{tet}_{1.5}(x)\) ; this is the same curve as in figure http://mizugadro.mydns.jp/t/index.php/File:Tetreal10bx10d.png

C++ generator of curves

/*Files GLxw2048.inc , TetSheldonIma.inc , ado.cin , conto.cin , filog.cin , fit1.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 std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
 #include "filog.cin"
z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
z_type a=log(b);
z_type Zo=Filog(a);
z_type Zc=conj(Filog(conj(a)));
DB A=32.;

z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
#include "GLxw2048.inc"
int K=2048;
//#include "ima6.inc"
#include "TetSheldonIma.inc"
z_type E[2048],G[2048];
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
c=0.;
//z+=z_type(    0.1196573712872846,     0.1299776198056910);
z+=z_type(      0.1196591376539 ,       0.1299777213955 );
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));} 
 cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
 cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
 c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
 return c;}

z_type TETB(z_type z){ int m,n; DB x=Re(z); 
if(x>.51) return exp(a*TETB(z-1.));
if(x<-.51) return log(TETB(z+1.))/a;
return tetb(z);
}

#include "fit1.cin"

int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;

FILE *o;
o=fopen("tetsheldor.eps","w");ado(o,1620,1320);
 fprintf(o,"210 610 translate\n 100 100 scale\n");

 for(m=-2;m<15;m++){if(m==0){M(m,-6.2)L(m,7.2)} else{M(m,-6)L(m,7)}}
 for(n=-7;n<8;n++){ M( -2,n)L(14,n)}
 fprintf(o,".008 W 0 0 0 RGB S\n");

DO(m,2410){x=-1.95+.01*m; z=z_type(x,0.);       

//      c=tetb(z); 
        c=TETB(z); 

        p=Re(c); q=Im(c); 
        y=p; if(m==0) M(x,y) else {if(y<20)L(x,y)} 
//      printf("%6.2lf %14.10lf %14.10lf\n",x,p,q); 
        if(x>14.||y>30.) break;
        }
 fprintf(o,".04 W 0 0 1 RGB S\n");

DO(m,2210){x=-1.99+ .01*m; z=z_type(x,0);       
//      c=tetb(z); 
        c=TETB(z); 
        p=Re(c); q=Im(c); 
        y=q; if(m==0) M(x,y) else {if(fabs(y)<20) L(x,y)} 
//      printf("%6.2lf %14.10lf %14.10lf\n",x,p,q); 
        if(x>14.|| p>1000.) break;
        }
 fprintf(o,".04 W 1 0 0 RGB S\n");

/* WOU! It works for complex values too! */
DO(m,2210){x=-1.95+ .01*m; z=z_type(x,0);       
//      c=tetb(z); 
//      c=TETB(z); 
//      c=FIT1(a,z); 
        c=FIT1(log(1.5),z); 
        p=Re(c); q=Im(c); 
        y=p; if(m==0) M(x,y) else {if(fabs(y)<20) L(x,y)} 
//      printf("%6.2lf %14.10lf %14.10lf\n",x,p,q); 
        if(x>14.|| p>10.) break;
        }
 fprintf(o,".01 W 0 0 0 RGB S\n");


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

c=TETB(0);
printf("tetb(0)= %16.14lf %16.14lf\n",Re(c),Im(c));

        system("epstopdf tetsheldor.eps"); 
        system( "open tetsheldor.pdf");
        getchar(); system("killall Preview");
 }

Latex generator of labels

\documentclass[12pt]{article}
 \usepackage{geometry}
 \paperwidth 1412pt
 \paperheight 1314pt
 \textwidth 2000pt
 \textheight 2000pt
 %\textwidth 700pt
 \usepackage{graphics}
% \usepackage{rotate}
\usepackage{rotating}
 \newcommand \rot {\begin{rotate}}
 \newcommand \ero {\end{rotate}}
 \newcommand \sx \scalebox
 \newcommand \ing \includegraphics
\parindent 0pt
\topmargin -92pt
\oddsidemargin -80pt
\begin{document}
\begin{picture}(1302,1304)
%\put(0,0){\ing{04}}
%\put(0,0){\ing{tetshelim}}
%\put(0,0){\ing{tetsheldore}}
\put(0,0){\ing{tetsheldor}}
\put(168,1286){\sx{6.7}{$y$}}
\put(170,1190){\sx{6}{$6$}}
\put(170,1090){\sx{6}{$5$}}
\put(170, 990){\sx{6}{$4$}}
\put(170, 890){\sx{6}{$3$}}
\put(170, 790){\sx{6}{$2$}}
\put(170, 690){\sx{6}{$1$}}
%\put(170, 590){\sx{6}{$0$}}
\put(120, 490){\sx{6}{$-1$}}
\put(120, 390){\sx{6}{$-2$}}
\put(120, 290){\sx{6}{$-3$}}
\put(120, 190){\sx{6}{$-4$}}
\put(120,  90){\sx{6}{$-5$}}

\put(60, 550){\sx{6}{$-1$}}
%\put(190, 550){\sx{6}{$0$}}
\put(294, 550){\sx{6}{$1$}}
\put(394, 550){\sx{6}{$2$}}
\put(494, 550){\sx{6}{$3$}}
\put(594, 550){\sx{6}{$4$}}
\put(694, 550){\sx{6}{$5$}}
\put(794, 550){\sx{6}{$6$}}
\put(894, 550){\sx{6}{$7$}}
\put(994, 550){\sx{6}{$8$}}
\put(1094, 550){\sx{6}{$9$}}
\put(1180, 550){\sx{6}{$10$}}
\put(1280, 550){\sx{6}{$11$}}
\put(1374, 550){\sx{6.7}{$x$}}

\put(330,800){\sx{7}{\rot{20}$y\!=\! \Re\big(\mathrm{tet}_b(x)\big)$\ero}}
\put(420,756){\sx{7}{\rot{10}$y\!=\! \mathrm{tet}_{1.5}(x)$\ero}}
\put(320,636){\sx{7}{\rot{6}$y\!=\! \Im\big(\mathrm{tet}_b(x)\big)$\ero}}
%\put(320,550){\sx{7}{\rot{2}$y\!=\! \Im\big(\mathrm{tet}_b(x)\big)$\ero}}
\end{picture}
\end{document}

References

Keywords

«[[]]», «Explicit plot», «Tetration to Sheldon base», «Superfunction», «Superfunctions», «Tetration»,

«Тетрация»,