File:Shelr80.png

Explicit plot of tetration to Sheldon base for real values of the argument.

Sheldon vase is defined with

$b= 1.52598338517+0.0178411853321 \,\mathrm i$.

Sheldon Levenstein has suggested this number, but he did not provide any way of evaluation of this number; so this value can be considered as exact.

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

Usage
This plot is prepared to use as figure 18.4 of the book Superfunctions

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:

// using namespace std; typedef std::complex z_type; 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.;
 * 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.)
 * 4) include "conto.cin"
 * 5) include "filog.cin"

z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd; int K=2048; //#include "ima6.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;}
 * 1) include "GLxw2048.inc"
 * 1) include "TetSheldonIma.inc"

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); }


 * 1) 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}