File:Penplot.jpg

Explicit plot of natural pentation,

$y=\mathrm{pen}(x)$ is shown with thick black line.

Other lines
The thin red line shows it asymptotic level $L\approx -1.8503545290271812$ is smallest real fixed point of natural tetration.

The thin blue line shows the asymptotic

$ y=L+\exp(k(x+x_1))$

where $k\approx 1.86573322821$

and $x_1 \approx 2.24817451898$

The thin green line shown the deviation from the linear approximation

$\mathrm{linear}(x)=1+x$

The deviation is denoted as $~\delta(x)=\mathrm{pen}(x)-\mathrm{linear}(x)$

In the range $-2.1\!<\!x\!<\!1.1$, the deviation is small, the linear approximation provides 2 correct significant digits. In order to make the deviation visible, it is scaled with factor 10, so, $y=10\delta(x)$ is plotted.

Description of natural pentation
Natiral pentation is specific superfunction of natural tetration (implementation fsexp.cin is available); so, the pentation safisfies the transfer equation

$\mathrm{pen}(z\!+\!1) = \mathrm{tet}\Big( \mathrm{pen}(z)\Big)$

The additional condition $\mathrm{pen}(1)=\mathrm e \approx 2.71$ is assumed.

Natural pentation is specific superfunction, it is constructed with regular iteration at the lowest real fixed point of natural tetration, denoted with $L$.

C++ generator of curves
Files ado.cin, fsexp.cin, fslog.cin should be loaded in order to compile the C++ code below typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x8.) return 999.; z=FSEXP(z);  if(abs(z)<40) goto L1; return 999.; L1: ;} return z; }

z_type pen(z_type z){ DB x; int m,n; x=Re(z); if(x<= -4.) return pen0(z); m=int(x+5.); z-=DB(m); z=pen0(z); DO(n,m) z=FSEXP(z); return z; }

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; FILE *o;o=fopen("penplo.eps","w"); ado(o,608,1008); fprintf(o,"404 204 translate\n 100 100 scale\n"); for(m=-4;m<3;m++) {M(m,-2)L(m,8)} for(n=-2;n<11;n++) {M( -4,n)L(2,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");
 * 1) define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);

DO(n,150){x=-4+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 0 RGB S\n");

DO(n,150){x=-2.2+.04*n;y=10.*(Re(pen7(x))-(1.+x)); if(n==0) M(x,y)else L(x,y); if(y>.3)break;} fprintf(o,".01 W 0 .5 0 RGB S\n");

DB L=-1.8503545290271812; DB K=1.86573322821; DB a=-.6263241; DB b=0.4827; DO(n,80){x=-4.+.04*n; DB e=exp(K*(x+2.24817451898));   y=L+e; if(n==0) M(x,y) else L(x,y); if(y>8.) break;} fprintf(o,".01 W 0 0 1 RGB S\n");

/* DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898));    y=L+e*(1.+e*(a)); if(n==0) M(x,y) else L(x,y); if(y>8.||y<-2.) break;} M(-4,L)L(0,L) fprintf(o,".01 W 1 0 0 RGB S\n");

/* DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898));    y=L+e*(1.+e*(a+e*b)); if(n==0) M(x,y) else L(x,y); if(y<-2.) break;} fprintf(o,".01 W 1 0 0 RGB S\n");

DB t2=M_PI/1.86573322821; DB tx=-2.32; fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);

printf("pen7(-1)=%18.14f\n", Re(pen7(-1.))); printf("Pi/1.86573322821=%18.14f %18.14f\n", M_PI/1.86573322821, 2*M_PI/1.86573322821);

system("epstopdf penplo.eps"); system(   "open penplo.pdf"); }

Latex generator of labels
\documentclass[12pt]{article} \paperwidth 608px \paperheight 1008px \textwidth 1394px \textheight 1300px \topmargin -104px \oddsidemargin -90px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document} {\begin{picture}(608,1008) %\put(12,0){\ing{24}} %\put(12,0){\ing{penma}} \put(0,0){\ing{penplo}} \put(377,994){\sx{3.2}{$y$}} \put(377,895){\sx{3.2}{$7$}} \put(377,795){\sx{3.2}{$6$}} \put(377,695){\sx{3.2}{$5$}} \put(377,594){\sx{3.2}{$4$}} \put(377,494){\sx{3.2}{$3$}} \put(377,394){\sx{3.2}{$2$}} \put(377,294){\sx{3.2}{$1$}} \put(377,194){\sx{3.2}{$0$}} \put(358, 93){\sx{3.2}{$-1$}} \put(80,174){\sx{3.2}{$-3$}} \put(180,174){\sx{3.2}{$-2$}} \put(280,174){\sx{3.2}{$-1$}} \put(396,174){\sx{3.2}{$0$}} \put(496,174){\sx{3.2}{$1$}} \put(590,174){\sx{3.2}{$x$}} \put(242,406){\sx{3.6}{\rot{85}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}} % %\put(560,510){\sx{3.6}{\rot{84}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}} %\put(532,708){\sx{3.6}{\rot{86}$y\!=\!\mathrm{pen}(x)$\ero}} \put(446,370){\sx{3.9}{\rot{70}$y\!=\!\mathrm{pen}(x)$\ero}} %\put(366,236){\sx{2.3}{$y\!=\!10(\mathrm{pen}(x)\!-\!1\!-\!x)$}} %\put(416,239){\sx{3.4}{$y=10\,\delta(x)$}} \put(8,236){\sx{3.3}{$y=10\,\delta(x)$}} %\put(366, 8){\sx{3.3}{$L$}} \put(312, 9){\sx{3.2}{$y\!=\!L$}} \end{picture} \end{document}