File:Fig1802.jpg
Original file (1,140 × 1,140 pixels, file size: 573 KB, MIME type: image/jpeg)
Summary
Testing of routine Conrec6.cin:
Reproduction of bottom part of Fig.18.2 at page 248 of book «Superfunctions» [1][2] with routine «Conrec6.cin» (instead of historic routine «conto.cin» [3] used at compilation of the book).
The short description of the figure,
and the generator of map (using «Conrec6.cin»),
and the historic generator of labels (in Latex, the same as in the book)
are copipasted below.
Description of function mapped
Complex map of function Filog is shown.
It is zoomin of the square of size unity from the central part of figure http://mizugadro.mydns.jp/t/index.php/File:Filogbigmap100.png
\[ u\!+\!\mathrm i v= \mathrm{Filog} (x\!+\!\mathrm i y) \]
Here \( z=\mathrm{Filog}(a) ~\) expresses solution \(z\) of equation \(\ln(z)/a=z\).
The two fixed points \(L\) of exponential to base \(b\) are expressed through function Filog as follows:
\[L=\mathrm{Filog}\big(\ln(b)\big)\]
and
\[L=\mathrm{Filog}\big(\ln(b^*)\big)^*\]
These fixed points are solutions of equation \[ \exp_b(L)=L \] These fixed points are important for construction of various superexponentials and abelexponentials, including tetration and arctetration.
In particular the map shows that for \(b=\sqrt{2}\), the fixed points are \(L=2\) and \(L=4\). The corresponding contours begin/end at the same point \(\ln\big(\sqrt{2}\big) \approx 0.346\) of the cut line \((-\infty, 1/\mathrm e)\approx (-\infty, 0.367)\).
In such a way, the same map shows the two fixed points of the exponential at once.
Also, the map shows that for base \(b>1/\mathrm e\), the exponential has no real fixed points.
The map is important as a reminder that not all holomorphic functions (used in the research) are real-holomorphic.
06.cc
// Files «ado.cin», «conrec6.cin», «filog.cin», «tania.cin» should be loaded in order to compile the C++ code below.
//If saved as «06.cc», the generator can be compiled with abracadabra «c++ -std=c++11 06.cc -O2 -o 06»
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
#include "filog.cin"
#include "Conrec6.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=500,M1=M+1;
int N=501,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("06.eps","w");ado(o,1020,1020);
fprintf(o,"510 510 translate\n 1000 1000 scale\n");
DO(m,M1) X[m]=-.5+.002*(m-.2);
DO(n,250)Y[n]=-.5+.002*n;
Y[250]=-.00001;
Y[251]= .00001;
for(n=252;n<N1;n++) Y[n]=-.5+.002*(n-1.);
for(m=-5;m<6;m++){M(.1*m,-.5)L(.1*m,.5)}
for(n=-5;n<6;n++){M( -.5,.1*n)L(.5,.1*n)}
fprintf(o,"2 setlinecap .001 W 0 0 0 RGB S\n");
M(exp(-1.),-.5)
L(exp(-1.), .5)
//M(M_PI/2.,-.5)
//L(M_PI/2., .5)
fprintf(o,".0007 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m+M1*n]=9999; f[m+M1*n]=9999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=Tania(z_type(-1.,-M_PI)+log(z))/(-z);
//c=Filog(z);
c=Filog(z);
p=Re(c);q=Im(c);
//if(p>-201. && p<201. && q>-201. && q<201. ){ g[m+N1*n]=p;f[m+N1*n]=q;}
if(p>-201. && p<201. && q>-201. && q<201. ){ g[m+M1*n]=p;f[m+M1*n]=q;}
}}
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=5.;q=1;
/*
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".0008 W 0 .6 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".0008 W .9 0 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".0008 W 0 0 .9 RGB S\n");
for(m=1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".002 W .9 0 0 RGB S\n");
for(m=1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".002 W 0 0 .9 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".002 W .6 0 .6 RGB S\n");
for(m=-20;m<21;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
p=10.;
conto(o,g,w,v,X,Y,M,N, 30 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, -50,-p,p); fprintf(o,".002 W .9 0 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, 50,-p,p); fprintf(o,".002 W 0 0 .9 RGB S\n");
*/
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)Conrec6(o,f,X,Y,M1,N1,(m+.1*n), q); fprintf(o,".0008 W 0 .6 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)Conrec6(o,g,X,Y,M1,N1,-(m+.1*n), q); fprintf(o,".0008 W .9 0 0 RGB S\n");
for(m=0;m<10;m++) for(n=2;n<10;n+=2)Conrec6(o,g,X,Y,M1,N1, (m+.1*n), q); fprintf(o,".0008 W 0 0 .9 RGB S\n");
for(m=1;m<20;m++) Conrec6(o,f,X,Y,M1,N1, (0.-m),p); fprintf(o,".002 W .9 0 0 RGB S\n");
for(m=1;m<20;m++) Conrec6(o,f,X,Y,M1,N1, (0.+m),p); fprintf(o,".002 W 0 0 .9 RGB S\n");
Conrec6(o,f,X,Y,M1,N1, (0. ),p); fprintf(o,".002 W .6 0 .6 RGB S\n");
for(m=-20;m<21;m++) Conrec6(o,g,X,Y,M1,N1, (0.+m),p); fprintf(o,".002 W 0 0 0 RGB S\n");
p=10.;
Conrec6(o,g,X,Y,M1,N1, 30 ,p); fprintf(o,".002 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 40 ,p); fprintf(o,".002 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 50 ,p); fprintf(o,".002 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1,-30 ,p); fprintf(o,".002 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1,-40 ,p); fprintf(o,".002 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1,-50 ,p); fprintf(o,".002 W 0 0 0 RGB S\n");
Conrec6(o,f,X,Y,M1,N1,-30 ,p); fprintf(o,".002 W .9 0 0 RGB S\n");
Conrec6(o,f,X,Y,M1,N1,-40 ,p); fprintf(o,".002 W .9 0 0 RGB S\n");
Conrec6(o,f,X,Y,M1,N1,-50 ,p); fprintf(o,".002 W .9 0 0 RGB S\n");
Conrec6(o,f,X,Y,M1,N1, 30 ,p); fprintf(o,".002 W 0 0 .9 RGB S\n");
Conrec6(o,f,X,Y,M1,N1, 40 ,p); fprintf(o,".002 W 0 0 .9 RGB S\n");
Conrec6(o,f,X,Y,M1,N1, 50 ,p); fprintf(o,".002 W 0 0 .9 RGB S\n");
//Conrec6(o,f,X,Y,M1,N1, 50,p); fprintf(o,".002 W 0 0 .9 RGB S\n");
//Conrec6(o,f,X,Y,M1,N1, 0,p); fprintf(o,".006 W .7 0 .7 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
system("epstopdf 06.eps");
system( "open 06.pdf"); //for mac
// getchar(); system("killall Preview"); // for mac
// Copyleft 2012-2025 by Dmitrii Kouznetsov
}
Generator of labels
% The input file «06.eps» should be prepared with the generator above in order to Latex the document below.
\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 1030pt
\paperheight 1030pt
\textwidth 1140pt
\textheight 1140pt
\topmargin -103pt
\oddsidemargin -86pt
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \rme {\mathrm{e}}
\newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(1020,1020)
%\put(10,10){\includegraphics{filogzo2}}
\put(10,10){\includegraphics{06}}
\put(0,1012){\sx{2.3}{$y$}}
\put(-7,914){\sx{2.3}{$\frac{4}{10}$}}
\put(-7,814){\sx{2.3}{$\frac{3}{10}$}}
\put(-7,714){\sx{2.3}{$\frac{2}{10}$}}
\put(-7,614){\sx{2.3}{$\frac{1}{10}$}}
\put( 2,514){\sx{2.3}{$0$}}
\put(-7,414){\sx{2.3}{$\frac{-\!1}{10}$}}
\put(-7,314){\sx{2.3}{$\frac{-\!1}{5}$}}
\put(-7,214){\sx{2.3}{$\frac{-\!3}{10}$}}
\put(-7,114){\sx{2.3}{$\frac{-\!2}{5}$}}
\put(-7, 14){\sx{2.3}{$\frac{-\!1}{2}$}}
\put(90, -1){\sx{2.3}{$-0.4$}}
\put(190, -1){\sx{2.3}{$-0.3$}}
\put(290, -1){\sx{2.3}{$-0.2$}}
\put(390, -1){\sx{2.3}{$-0.1$}}
\put(516, -1){\sx{2.3}{$0$}}
\put(609, -1){\sx{2.3}{$0.1$}}
\put(709, -1){\sx{2.3}{$0.2$}}
\put(809, -1){\sx{2.3}{$0.3$}}
\put(866, -1){\sx{2.2}{$1/\mathrm e$}}
\put(910, -1){\sx{2.3}{$0.4$}}
\put(1010, 0){\sx{2.4}{$x$}}
\put(250,540){\sx{3}{\rot{82}$u\!=\!0.8$ \ero} }
\put(530,540){\sx{3}{\rot{81}$u\!=\!1$ \ero} }
\put(680,524){\sx{2.8}{\rot{77}$u\!=\!1.2$ \ero} }
\put(768,524){\sx{2.8}{\rot{72}$u\!=\!1.4$ \ero} }
\put(816,524){\sx{2.8}{\rot{56}$u\!=\!1.6$ \ero} }
\put(600,684){\sx{2.8}{\rot{-33}$v\!=\!0.2$ \ero} }
\put(720,798){\sx{2.8}{\rot{-62}$v\!=\!0.4$ \ero} }
\put(820,490){\sx{3}{\rot{-4}$v\!=\!1$ \ero} }
\put(820,466){\sx{3}{\rot{-12}$v\!=\!2$ \ero} }
\put(808,446){\sx{3}{\rot{-26}$v\!=\!3$ \ero} }
\put(768,372){\sx{3}{\rot{36}$u\!=\!5$ \ero} }
\put(770,350){\sx{3}{\rot{26}$u\!=\!4$ \ero} }
\put(770,324){\sx{3}{\rot{16}$u\!=\!3$ \ero} }
\put(768,297){\sx{3}{\rot{2}$u\!=\!2$ \ero} }
\put(766,262){\sx{3}{\rot{-8}$u\!=\!1$ \ero} }
\put(752,230){\sx{3}{\rot{-28}$u\!=\!0$ \ero} }
\put(94,99){\sx{3}{\rot{41}$v\!=\!-2$ \ero} }
\put(162,50){\sx{3}{\rot{57}$v\!=\!-1$ \ero} }
\put(242,30){\sx{3}{\rot{75}$v\!=\!0$ \ero} }
\put(315,30){\sx{3}{\rot{87}$v\!=\!1$ \ero} }
\end{picture}
\end{document}
Warning
At the moment of loading this image, the routine Conrec6.cin is under testing.
However, you may load the preliminary version and use it at your own risk.
At the reuse, please, attribute the source and indicate the modification(s) if any.
This helps to trace errors, mistakes, misprints if any.
Copyleft 2025 by Dmitrii Kouznetsov and ChatGPT.
References
- ↑ https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 Superfunctions // Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics // 978-620-2-67286-3 // Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. .. // Authors Dmitrii Kouznetsov // Book language English // Published on 2020-07-28 // Publishing house LAP LAMBERT Academic Publishing // Number of pages 328 // Price (EUR ) €52.90 // (as it is seen from year 2025)
- ↑ https://mizugadro.mydns.jp/BOOK/468.pdf D.Kouznetsov. Superfunctions. 2015-2020.
- ↑ https://en.citizendium.org/wiki/Contour_plot/Code/conto.cin // Copyleft 2008 by Dmitrii Kouznetsov This page was last modified 01:05, 2 February 2009.
Keywords
«Book», «C++», «ChatGPT», «Complex map», «Conrec6.cin», «Exp», «Filog», «Filog.cin», «Fixed point», «Latex», «Logarithm», «Superfunctions», «Tania.cin», «Tetration», «Zoom»,
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 10:39, 6 September 2025 | | 1,140 × 1,140 (573 KB) | T (talk | contribs) | == Summary == {{oq|Fig1802.jpg|Original file (1,140 × 1,140 pixels, file size: 573 KB, MIME type: image/jpeg) }} Testing of routine Conrec6.cin: Reproduction of bottom part of Fig.18.2 at page 248 of book «Superfunctions» <ref> https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 Superfunctions // Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics // 978-620-2-67286-3 // Tools for evaluation of superfunc... |
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