File:Sqrt2eitet.jpg

iterates of the esponent to base $\sqrt{2}$, constructed with tetration and arctetration to thie base.

Usage: this is figure 16.7 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

Evaluation of tetration and arvtetration to base \sqrt{2}$ is described also in article .

C++ generator of the curves
Files ado.cin, sqrt2f21e.cin, sqrt2f21l.cin should be loaded in order to compile the code below. typedef std::complex z_type; // #include "sqrt2f45e.cin" // #include "sqrt2f45l.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);}
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"
 * 1) include "sqrt2f21e.cin"
 * 2) include "sqrt2f21l.cin"

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"); 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"); }
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

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}