File:TetKK200.png

Parameters, that determine the asymptotic behavior of Tetration to real base \(b\), versus logarithm of this base, \( \beta=\ln(b)\).

These parameters are:

Fixed point \(L=\) Filog\((\beta)\)

Asymptoitic growing factor \( K= \beta L \)

Asymptotic increment \( k= \ln(K) \)

Values correspond to the upped half of the complex plane; so, \(\Im(L)\ge 0\). Curve for \(L^*\) is not drawn.

Requirements

For generation of the image, the following files should be loaded:

ado.cin

Filog.cin

k12.cc and TetKK.tex below.

k12.cc

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex>
#define z_type std::complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "Filog.cin"
#include "../ado.cin"

int main(){  z_type b, beta, L,K,k; int M,m,n; DB x,y;
FILE *o=fopen("k12.eps","w");
#define M(x,y) fprintf(o,"%7.4lf %7.4lf M\n", 0.+x, 0.+y);
#define L(x,y) fprintf(o,"%7.4lf %7.4lf L\n", 0.+x, 0.+y);
ado(o,322,522);
fprintf(o,"10 110 translate 100 100 scale 2 setlinecap 1 setlinejoin\n");
for(m=0;m<4;m++){M(m,0)L(m,4)}
for(n=0;n<5;n++){M(0,n)L(3,n)}
fprintf(o,".006 W S\n");
M(exp(-1.),0)L(exp(-1.),4) fprintf(o,".002 W S\n");
M(log(2.)/2.,0)L(log(2.)/2.,4) fprintf(o,".002 W S\n");
DB x0=M_PI_2; M(x0,0)L(x0,1) fprintf(o,".002 W S\n");
//for(n=0;n<31;n++){ beta=.1*n+1.e-15*I; x=Re(beta); y=Re(exp(beta));if(n==0) M(x,y) else L(x,y) } fprintf(o,".004 W S\n");
DO(n,610){beta=.006+.005*n-1.e-15*I;L=Filog(beta); x=Re(beta); y=Re(L);if(n==0)M(x,y)else L(x,y)} fprintf(o,"0 0 1 RGB .005 W S\n");
DO(n,610){beta=.006+.005*n+1.e-15*I;L=Filog(beta); x=Re(beta); y=Re(L);if(n==0)M(x,y)else L(x,y)} fprintf(o,"0 0 1 RGB .005 W S\n");
DO(n,610){beta=.006+.005*n+1.e-15*I;L=Filog(beta); x=Re(beta); y=Im(L);if(n==0)M(x,y)else L(x,y)} fprintf(o,"1 0 0 RGB .005 W S\n");
DO(n,610){beta=.004+.005*n-1.e-15*I;L=Filog(beta); K=beta*L; x=Re(beta);y=Re(K);if(n==0)M(x,y)else L(x,y)} fprintf(o,"0 0 .9 RGB .018 W S\n");
DO(n,610){beta=.004+.005*n+1.e-15*I;L=Filog(beta); K=beta*L; x=Re(beta);y=Re(K);if(n==0)M(x,y)else L(x,y)} fprintf(o,"0 0 1 RGB .02 W S\n");
DO(n,610){beta=.006+.005*n-1.e-15*I;L=Filog(beta); K=beta*L; x=Re(beta);y=Im(K);if(n==0)M(x,y)else L(x,y)} fprintf(o,"1 0 0 RGB .02 W S\n");
DO(n,610){beta=.002+.005*n+1.e-15*I;L=Filog(beta); k=log(beta*L); x=Re(beta);y=Re(k);if(n==0)M(x,y)else L(x,y)} fprintf(o,".7 0 .7 RGB .007 W S\n");
DO(n,610){beta=.002+.005*n-1.e-15*I;L=Filog(beta); k=log(beta*L); x=Re(beta);y=Re(k);if(n==0)M(x,y)else L(x,y)} fprintf(o,".7 0 .7 RGB .007 W S\n");
DO(n,610){beta=.002+.005*n+1.e-15*I;L=Filog(beta); k=log(beta*L); x=Re(beta);y=Im(k);if(n==0)M(x,y)else L(x,y)} fprintf(o,"0 0 0 RGB .007 W S\n");
fprintf(o,"\%\%Showpage trailer\n");
fclose(o);
system("epstopdf k12.eps");
system("    xpdf k12.pdf");
}

TetKK.tex

 \documentclass[12pt]{article}
 \usepackage{geometry}
 \paperwidth 324pt
 \paperheight 462pt
 \usepackage{graphicx}
 \newcommand \sx {\scalebox}
 \newcommand \ing {\includegraphics}
% \usepackage{rotate}
 \usepackage{rotating}
 \newcommand \rot {\begin{rotate}}
 \newcommand \ero {\end{rotate}}
 \topmargin -108pt
 \oddsidemargin -72pt
 \parindent 0pt
 \begin{document}
 \begin{picture}(320,530)
%\put(5,10){\ing{k03.pdf}}
 \put(5,10){\ing{k12.pdf}}
 \put(0,520){\sx{2}{$y$}}
 \put(0,413){\sx{2}{$3$}}
 \put(0,313){\sx{2}{$2$}}
 \put(0,213){\sx{2}{$1$}}
 \put(0,113){\sx{2}{$0$}}
 \put(10,100){\sx{2}{$0$}}
 \put(110,100){\sx{2}{$1$}}
 \put(165,101){\sx{2}{$\frac{\pi}{2}$}}
 \put(210,100){\sx{2}{$2$}}
% \put(110,300){\sx{2}{$3$}}
 \put(308,100){\sx{2.1}{$\beta$}}
 \put(56,424){\rot{90}\sx{1.6}{$y\!=\!\Re(L)$}\ero}
 \put(27,406){\rot{-85}\sx{1.6}{$y\!=\!\Re(K)$}\ero}
 \put(194,292){\rot{9}\sx{1.6}{$y\!=\!\Im(K)$}\ero}
 \put(198,270){\rot{8}\sx{1.6}{$y\!=\!\Im(k)$}\ero}
 \put(194,206){\rot{-17}\sx{1.6}{$y\!=\!\Im(L)$}\ero}
 \put(188,163){\rot{8}\sx{1.6}{$y\!=\!\Re(k)$}\ero}
 \put(226,94){\rot{-17}\sx{1.6}{$y\!=\!\Re(K)$}\ero}
 \end{picture}
 \end{document}

Generation command

make k11
./k11
Latex TetKK
convert -density 200 TetKK.pdf PNG8:TetKK200.png

References