File:Ackerplot400.jpg

Summary

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Description

${\displaystyle A_{\mathrm {e} ,1}=z\mapsto b+z}$

${\displaystyle A_{\mathrm {e} ,2}=z\mapsto b\,z}$

${\displaystyle A_{\mathrm {e} ,3}=\exp }$

${\displaystyle A_{\mathrm {e} ,4}=\mathrm {tet} }$

${\displaystyle A_{\mathrm {e} ,5}=\mathrm {pen} }$

The ackermanns ${\displaystyle A}$ satisfy equations

${\displaystyle A_{b,1}(z)=b+z}$

${\displaystyle A_{b,m}(1)=b}$ for ${\displaystyle m>1}$

${\displaystyle A_{b,m+1}(z\!+\!1)=A_{m}(A(m\!+\!1,z))}$ for ${\displaystyle m>1}$

The additional requirements are applied for the uniqueness.

In particular, for the real ${\displaystyle b}$, the real holomorphism is assumed, ${\displaystyle A_{b,1}(z^{*})=A_{b,1}(z^{*})}$.

C++ generator of curves

Files fsexp.cin and fslog.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.) 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 */ #include "fsexp.cin" #include "fslog.cin" z_type pen0(z_type z){ DB Lp=-1.8503545290271812; DB k,a,b; k=1.86573322821; a=-.6263241; b=0.4827; z_type e=exp(k*z); return Lp + e*(1.+e*(a+b*e)); } z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.)); DO(n,7) { if(Re(z)>8.) 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,608,808); fprintf(o,"304 204 translate\n 100 100 scale\n"); #define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y); #define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y); #define o(x,y) fprintf(o,"%8.4f %8.4f o\n",0.+x,0.+y); for(m=-3;m<4;m++) {M(m,-2)L(m,6)} for(n=-2;n<9;n++) {M( -3,n)L(3,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n"); M(-3.02,-3.02+M_E)L(3.02,3.02+M_E) fprintf(o,".007 W .3 0 .3 RGB S\n"); M(-1., -M_E)L(3.02,3.02*M_E) fprintf(o,".007 W 0 .5 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>6.)break;} fprintf(o,".02 W 0 .8 0 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>6.)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>6.)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"); 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 604px \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(496,174){\sx{3.}{$2$}} \put(586,174){\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(504,712){\sx{1.8}{\rot{70}$y\!=\!\mathrm{e}x$\ero}} \put(538,718){\sx{1.96}{\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}

References

D.Kouznetsov. Holomorphic ackermanns. 2014-2015, in preparation.