File:TetPlotU.png

Explicit plot of tetration to base e; $y=\mathrm{tet}(x)$ is shown with thick pink line.

For comparison, the thin black line shows the exponential, $y=\exp(x)$

For $x$ between $-1$ and $0$, the graphic of tetration $y\!=\!\mathrm{tet}(x)$ looks similar to that of linear function $y\!=\!x\!+\!1$. The difference between these two functions, scaled with factor 10, id est, $y=10\Big( \mathrm{tet}(x)-(x+1)\Big)$, is plotted with blue curve of intermediate thickness. (It would be difficult to see the difference without scaling).

Between $0$ and $1$, the graphic of tetration $y\!=\!\mathrm{tet}(x)$ looks similar to that of the exponential function $y\!=\!\exp(x)$. These curves cross three times, at $x\!=\!0$, at $x\!=\!x_{\rm half}\!\approx\! 0.47$ and at $x\!=\!1$.

Correspondently, the thick pink curve for $y\!=\!\mathrm{tet}(x)$ woud cross the graphic $~y=x\!+\!1~$ at $x\!=\!x_{\rm half}\!-\!1\!\approx\!-0.53$, and at this value, the difference $\mathrm{tet}(x) - (x\!+\!1)$ becomes zero.

C++ generator of curves
// files fsexp.cin, fslog.cin and ado.cin should be loaded to the working directory in order to compile the C++ code below:

using namespace std; typedef complex z_type; main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=400,M1=M+1; int N=401,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array FILE *o;o=fopen("TetPlot.eps","w");ado(o,402,1002); fprintf(o,"201 201 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(m=-2;m<3;m++){M(m,-2)L(m,8)} for(n=-2;n<11;n++){M(-2,n)L(2,n)} fprintf(o,".004 W S\n"); DO(m,101){y=-2.+.1*m; x=Re(FSLOG(y)); printf("%4.3f %4.3f \n",x,y); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W 1 0 1 RGB S\n"); DO(m,44){x=-2.+.1*m; y=exp(x); printf("%4.3f %4.3f \n",x,y); if(m==0)M(x,y) else L(x,y);} fprintf(o,".008 W 0 0 0 RGB S\n"); DO(m,161){x=-1.3+.01*m; y=Re(FSEXP(x))-(1.+x); y*=10; printf("%4.3f %4.3f \n",x,y); if(m==0)M(x,y) else L(x,y);} fprintf(o,".02 W 0 0 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf TetPlot.eps"); system(   "open TetPlot.pdf"); //for mac getchar; system("killall Preview"); // for mac } // Copyleft 2012 by Dmitrii Kouznetsov
 * 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 "ado.cin"
 * 5) include "fslog.cin"
 * 6) include "fsexp.cin"
 * 1) define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);

Latex benerator of labels
% % \documentclass[12pt]{article} % \paperheight 1002px % \paperwidth 402px % \textwidth 1294px % \textheight 1100px % \topmargin -105px % \oddsidemargin -72px % \usepackage{graphics} % \usepackage{rotating} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \newcommand \rmi {\mathrm{i}} % \begin{document} % \newcommand \zoomax { % \put(184,989){\sx{2.8}{$y$}} % \put(184, 893){\sx{2.6}{$7$}} % \put(184, 793){\sx{2.6}{$6$}} % \put(184, 693){\sx{2.6}{$5$}} % \put(184, 593){\sx{2.6}{$4$}} % \put(184, 493){\sx{2.6}{$3$}} % \put(184, 393){\sx{2.6}{$2$}} % \put(184, 293){\sx{2.6}{$1$}} % \put(184, 193){\sx{2.6}{$0$}} % \put(170, 092){\sx{2.6}{$-\!1$}} % %\put(-1, 010){\sx{2.6}{$-\!2$}} % %\put(016, -4){\sx{2.6}{$-\!2$}} % \put(080,176){\sx{2.6}{$-\!1$}} % \put(195,176){\sx{2.6}{$0$}} % \put(295,176){\sx{2.6}{$1$}} % %\put(435, -5){\sx{3}{$2$}} % \put(386,179){\sx{2.7}{$x$}} % } % \parindent 0pt % \sx{1}{\begin{picture}(852,1002) % %\put(40,20){\ing{b271tMap3}} % \zoomax % \put(0,0){\ing{TetPlot}} % \put(222,642){\sx{2.5}{$y\!=\!\mathrm{tet}(x)$}} % \put(024,44){\sx{2.5}{$y\!=\!\mathrm{tet}(x)$}} % \put(334,572){\sx{2.5}{$y\!=\!\mathrm e^x$}} % \put(010,232){\sx{2.5}{$y\!=\!\mathrm e^x$}} % \put(204,259){\sx{1.7}{$y\!=\!10\Big(\mathrm{tet}(x)-(x\!+\!1)\Big)$}} % \put(72,130){\sx{1.9}{$y\!=\!10\Big(\mathrm{tet}(x)-(x\!+\!1)\Big)$}} % \end{picture}} % \end{document} %