File:Ackerplot.jpg

Explicit plot of the 5 Ackermann functions to base $\mathrm e$

$y=A_{\mathrm e,n}(x)~$ for $n=1,2,3,4,5$

$A_{\mathrm e,1}(z)=\mathrm e + z$

$A_{\mathrm e,2}(z)=\mathrm e \,z$

$A_{\mathrm e,3}(z)=\mathrm e^z~$, see exp

$A_{\mathrm e,4}(z)=\mathrm{tet}(z)~$, see natural tetration

$A_{\mathrm e,5}(z)=\mathrm{pen}(z)~$, see natural pentation

General properties of Ackermann functions
The Ackermann function, or ackermann, to base $b$ satisfies the transfer equation

$A_{b,n}(z\!+\!1)=A_{b,n-1}\big( A_{b,n}(z) \Big)$

and the "initial" conditions $~A_{b,1}(z)\!=\!b\!+\!z~$, $A_{b,n}(1)\!=\!b$ for $n\!>\!1$

These conditions are not sufficient to provide the uniqueness of the ackermanns. The holomorphism with respect to the argument should be also required.

$n$ is interpreted as number of the ackermann; is supposed to be natural number.

For real base $b$, the ackermann is real-holomorphic, $A_{b,n}(z^*)=A_{b,n}(z)^*$.

C++ generator of curves
Files fsexp.cin and fslog.cin should be loaded in order to compile the code below. typedef std::complex z_type; // #include "ado.cin" void ado(FILE *O, int X, int Y) {      fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%'); fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y); fprintf(O,"/M {moveto} bind def\n"); fprintf(O,"/L {lineto} bind def\n"); fprintf(O,"/S {stroke} bind def\n"); fprintf(O,"/s {show newpath} bind def\n"); fprintf(O,"/C {closepath} bind def\n"); fprintf(O,"/F {fill} bind def\n"); fprintf(O,"/o {.01 0 360 arc C F} bind def\n"); fprintf(O,"/times-Roman findfont 20 scalefont setfont\n"); fprintf(O,"/W {setlinewidth} bind def\n"); fprintf(O,"/RGB {setrgbcolor} bind def\n");} /* end of routine */
 * 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("ackerplo.eps","w"); ado(o,508,808); fprintf(o,"304 204 translate\n 100 100 scale\n"); for(m=-3;m<3;m++) {M(m,-2)L(m,6)} for(n=-2;n<9;n++) {M( -3,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);
 * 3) define o(x,y) fprintf(o,"%8.4f %8.4f o\n",0.+x,0.+y);

M(-3.02,-3.02+M_E)L(2.02,2.02+M_E) fprintf(o,".007 W .3 0 .3 RGB S\n"); M(-1., -M_E)L(2.02,2.02*M_E) fprintf(o,".007 W 0 .5 0 RGB S\n");

//DO(n,156){x=-4+.04*n;y=exp(x); if(n==0) M(x,y)else L(x,y); if(y>7.)break;} fprintf(o,".02 W 0 .8 0 RGB S\n"); fprintf(o,"1 0 0 RGB\n"); DO(n,306){x=-3.02+.02*(n-.5);y=exp(x); o(x,y); if(y>7.)break;} fprintf(o,".02 W 0 .8 0 RGB S\n");

//DO(n,150){x=-1.9+.04*n;y=Re(FSEXP(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 1 RGB S\n"); DO(n,202){y=-3+.05*(n-.6);x=Re(FSLOG(y)); if(n/2*2==n) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,"0 setlinecap .016 W 0 0 1 RGB S\n");

DO(n,150){x=-3.03+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>7.)break;} fprintf(o,".01 W 0 0 0 RGB S\n");

DB L=-1.8503545290271812; DB K=1.86573322821; DB a=-.6263241; DB b=0.4827; M(-3,L)L(0,L) M(0,M_E) L(1,M_E) fprintf(o,".002 W 0 0 0 RGB S\n"); //M(-4.1,-4.1+M_E) L(4.1,4.1+M_E) fprintf(o,".002 W 0 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 ackerplo.eps"); system(   "open ackerplo.pdf"); }

Latex generator of curves
\documentclass[12pt]{article} \paperwidth 504px \paperheight 806px \textwidth 1394px \textheight 1300px \topmargin -104px \oddsidemargin -92px \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}(608,806) %\put(12,0){\ing{penma}} \put(0,0){\ing{ackerplo}} \put(277,788){\sx{3.}{$y$}} \put(277,695){\sx{3.}{$5$}} \put(277,594){\sx{3.}{$4$}} \put(277,494){\sx{3.}{$3$}} \put(278,468){\sx{3.}{$\mathrm e$}} \put(277,394){\sx{3.}{$2$}} \put(277,294){\sx{3.}{$1$}} \put(277,194){\sx{3.}{$0$}} \put(258, 93){\sx{3.}{$-1$}} \put( 80,174){\sx{3.}{$-2$}} \put(180,174){\sx{3.}{$-1$}} \put(296,174){\sx{3.}{$0$}} \put(396,174){\sx{3.}{$1$}} \put(486,174){\sx{3.}{$x$}} \put(438,714){\sx{1.8}{\rot{85}$y\!=\!\mathrm{pen}(x)$\ero}} \put(460,716){\sx{1.8}{\rot{82}$y\!=\!\mathrm{tet}(x)$\ero}} \put(478,712){\sx{1.8}{\rot{77}$y\!=\!\mathrm{exp}(x)$\ero}} \put(488,686){\sx{1.8}{\rot{70}$y\!=\!\mathrm{e}x$\ero}} %\put(478,628){\sx{1.8}{\rot{50}$y\!=\!\mathrm{e}\!+\!x$\ero}} \put(338,518){\sx{1.9}{\rot{44}$y\!=\!\mathrm{e}\!+\!x$\ero}} % \put(86,222){\sx{1.9}{\rot{11}$y\!=\!\mathrm{exp}(x)$\ero}} \put(20,30){\sx{1.9}{\rot{30}$y\!=\!\mathrm{pen}(x)$\ero}} \put(138,22){\sx{1.9}{\rot{74}$y\!=\!\mathrm{tet}(x)$\ero}} \put(252,22){\sx{1.9}{\rot{70}$y\!=\!\mathrm{e} x$\ero}} % \put(308, 13){\sx{2.2}{$y\!=\!L_{\mathrm e,4,0}$}} \end{picture} \end{document}