File:Ackerplot400.jpg

Summary

Licensing

   

This media, Ackerplot400.jpg, is licenced under the Creative Commons Attribution 3.0 Unported License

You are free: To Share — To copy, distribute and transmit the work; To Remix — To adapt the work.
Under the following conditions: Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
For any reuse or distribution, you must make clear to others the licence terms of this work (the best way to do this is with a link to this licence's web page). Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this licence impairs or restricts the author's moral rights.
Read the full licence.

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 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" 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.