File:Sqrt2eitet.jpg

Fig.16.6 from page 228 of book Superfunctions^[1], 2020.

This image appears also as Рис.16.7 at page 232 of the Russian version «Суперфункции»^[2], 2014.

The picture shows iterates of the exponential to base \(\sqrt{2}\):

\( y=\exp_2^{\ n}(x) \)

for various values of number \(n\).

For non-integer values of \(n\), the iterates are expressed trough the tetration \(\mathrm{tet}_2\) and the arctetration \(\mathrm{ate}_2\):

\( \exp_2^{\ n}(z)= \mathrm{tet}_2( n + \mathrm{ate}_2(z) ) \)

Evaluation of tetration and arctetration to base \(\sqrt{2}\) is described also in Mathematics of Computation ^[3], 2010.

C++ generator of the curves

/* Files ado.cin, sqrt2f21e.cin, sqrt2f21l.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++)
 #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 "ado.cin"
// #include "sqrt2f45e.cin"
// #include "sqrt2f45l.cin"
 #include "sqrt2f21e.cin"
 #include "sqrt2f21l.cin"
 DB B=sqrt(2.);
 DB F(DB z) { return exp( exp( log(B)*z));}
 DB G(DB z) { return log( log(z) )/log(B);}

int main(){ int m,n; double x,y,t; FILE *o;
 o=fopen("itereq2tlo.eps","w"); ado(o,1420,1420);
 fprintf(o,"701 701 translate 100 100 scale\n");
 #define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
M(-7,1.99)L(3.995,2.01)L(4.02,7)
fprintf(o,"1 setlinecap 1 setlinejoin .03 W 0 .5 1 RGB S\n");
M(1.99,-7)L(2.01,3.995)L(7,4.02)
fprintf(o,"1 setlinecap 1 setlinejoin .03 W 1 .5 0 RGB S\n");
M(-7,-7)L(7,7)
fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
 for(m=-7;m<8;m++) {M(m,-7)L(m,7)}
 for(m=-7;m<8;m++) {M(-7,m)L(7,m)}
 fprintf(o,"2 setlinecap .01 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinecap 1 setlinejoin\n");
DO(m,141){x=-7.01+.1*m;y=exp(log(B)*x);y=exp(log(B)*y);y=exp(log(B)*y); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y); if(y>7.) break;}
DO(m,141){x=-7.01+.1*m;y=exp(log(B)*x);y=exp(log(B)*y);y=exp(log(B)*y); if(m==0) M(x,y) else L(x,y);if(y>7.) break;} 
fprintf(o,".04 W 0 0 1 RGB S\n");
DO(m,141){x=-7.01+.1*m; y=exp(log(B)*x); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y); if(y>7.) break; }
DO(m,141){x=-7.01+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y); if(y>7.) break;} 
fprintf(o,".04 W 0 .5 1 RGB S\n");
DO(m,71){x=.01+.1*m; y=log(x)/log(B); if(m==0)M(x,y) else L(x,y); if(y>7.) break; } 
fprintf(o,".04 W 1 .5 0 RGB S\n");
DO(m,141){x=-7.01+.1*m;y=exp(log(B)*x);y=exp(log(B)*y);y=exp(log(B)*y); y=exp(log(B)*y); if(m==0)M(y,x) else L(y,x); if(y>7.) break;}
DO(m,141){x=-7.01+.1*m;y=exp(log(B)*x);y=exp(log(B)*y);y=exp(log(B)*y); if(m==0) M(y,x) else L(y,x);if(y>7.) break;} 
fprintf(o,".04 W 1 0 0 RGB S\n");
DO(m,141){x=-7.01+.1*m; y=exp(log(B)*x); y=exp(log(B)*y); if(m==0)M(y,x) else L(y,x); if(y>7.) break; }
DO(m,141){x=-7.01+.1*m; y=exp(log(B)*x); if(m==0)M(y,x) else L(y,x); if(y>7.) break;} 
fprintf(o,".04 W 1 .5 0 RGB S\n");
/*
DO(m,131){x=1.41+.1*m;y=log(x)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);} 
DO(m,131){x=1.63+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);} 
DO(m,131){x=1.75+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);} 
*/
fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 .5 0 RGB S\n");
// for(n=-20;n<21;n++){t=.1*n; M(2,2); DO(m,122){x=2.05+.1*m; y=Re(F45E(t+F45L(x+1.e-14*I))); L(x,y); if(y>14.1)break;} }
for(n=-20;n<21;n++){t=.1*n; M(4,4); DO(m,221){x=3.95-.05*m; y=Re(F21E(t+F21L(x+1.e-12*I))); L(x,y); if(y>14.1 || y<-7.)break;} }
fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf itereq2tlo.eps");    
       system(    "open itereq2tlo.pdf");
       getchar(); system("killall Preview");
 }

Latex generator of the labels

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 1470pt 
\paperheight 1456pt 
\topmargin -103pt 
\oddsidemargin -52pt 
\textwidth 1604pt 
\textheight 1600pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\parindent 0pt
\pagestyle{empty} 
\begin{document} 
\begin{picture}(1446,1446) 
%\put(10,10){\ing{IterPowPlot}} 
\put(40,40){\ing{Itereq2tlo}} 
\put(4,1420){\sx{4.4}{$y$}} 
\put(04,1333){\sx{4}{$6$}} 
\put(04,1233){\sx{4}{$5$}} 
\put(04,1133){\sx{4}{$4$}} 
\put(04,1033){\sx{4}{$3$}} 
\put(04, 933){\sx{4}{$2$}} 
\put(04, 833){\sx{4}{$1$}} 
\put(04, 733){\sx{4}{$0$}} 
\put(-24, 632){\sx{4}{$-1$}} 
\put(-24, 532){\sx{4}{$-2$}} 
\put(-24, 432){\sx{4}{$-3$}} 
\put(-24, 332){\sx{4}{$-4$}} 
\put(-24, 232){\sx{4}{$-5$}} 
\put(-24, 132){\sx{4}{$-6$}} 
 
\put(100,0){\sx{4}{$-6$}} 
\put(200,0){\sx{4}{$-5$}} 
\put(300,0){\sx{4}{$-4$}} 
\put(400,0){\sx{4}{$-3$}} 
\put(500,0){\sx{4}{$-2$}} 
\put(600,0){\sx{4}{$-1$}} 
\put(730,0){\sx{4}{$0$}} 
\put(830,0){\sx{4}{$1$}} 
\put(930,0){\sx{4}{$2$}} 
\put(1030,0){\sx{4}{$3$}} 
\put(1130,0){\sx{4}{$4$}} 
\put(1230,0){\sx{4}{$5$}} 
\put(1330,0){\sx{4}{$6$}} 
\put(1422,0){\sx{4}{$x$}} 

\put( 66,930){\sx{5.5}{\rot{0}$n\!\rightarrow + \infty$\ero}}
\put( 66,830){\sx{5.5}{\rot{1}$n\!=\!2$\ero}}
\put( 66,736){\sx{5.5}{\rot{3}$n\!=\!1$\ero}}
\put( 78,618){\sx{5.5}{\rot{5}$n\!=\!0.5$\ero}}
\put( 96,522){\sx{5.5}{\rot{11}$n\!=\!0.3$\ero}}
\put(116,448){\sx{5.5}{\rot{16}$n\!=\!0.2$\ero}}
\put(142,344){\sx{5.5}{\rot{25}$n\!=\!0.1$\ero}}
\put(212,190){\sx{5.8}{\rot{44}$n\!=\!0$\ero}}
\put(362,100){\sx{5.5}{\rot{64}$n\!=\!-0.1$\ero}}
\put(470, 60){\sx{5.5}{\rot{73}$n\!=\!-0.2$\ero}}
\put(606, 50){\sx{5.5}{\rot{82}$n\!=\!-0.4$\ero}}
\put(770, 50){\sx{5.5}{\rot{86}$n\!=\!-1$\ero}}
\put(866, 50){\sx{5.5}{\rot{88}$n\!=\!-2$\ero}}
\put(964, 50){\sx{5.5}{\rot{90}$n\!\rightarrow -\infty$\ero}}
%
\put(1222,1298){\sx{5.5}{\rot{74}$n\!=\!2$\ero}}
\put(1255,1288){\sx{5.5}{\rot{64}$n\!=\!1$\ero}}
\put(1302,1282){\sx{5.5}{\rot{44}$n\!=\!0$\ero}}
\put(1272,1208){\sx{5.5}{\rot{24}$n\!=\!-1$\ero}}
\end{picture} 
\end{document} 

Refereces

Keywords

«[[]]»,

«Arctetration», «Base sqrt2», «Exponential», «Iterte», «Superfunction», «Superfunctions», «Tetration»,

«[[]]», «Суперфункции», «[[]]»,