Difference between revisions of "File:Filogbigmap100.png"

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(Importing image file)
 
($ -> \( ; description ; refs ; pre ; keywords)
 
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  +
{{oq|Filogbigmap100.png|Original file ‎(2,870 × 2,851 pixels, file size: 847 KB, MIME type: image/png)|800}}
−
Importing image file
  
  +
  +
{{fig|Filogmap300.png|400|-2|4|4|The zoom-in of the central part of the map}}
  +
  +
Fig.18.1 from page 247 of book «[[Superfunctions]]»
  +
<ref name="be">
  +
https://mizugadro.mydns.jp/BOOK/468.pdf
  +
D.Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>, 2020.
  +
  +
The same picture appears also as Рис.18.1 at page 254 of the Russian version «[[Суперфункции]]»
  +
<ref name="br">
  +
https://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
  +
</ref>, 2014.
  +
  +
The figure shows the [[Complex map]] of function [[Filog]].
  +
  +
The zooming of the central part of this map is shown in figure at tight.
  +
  +
==Semantics of Filog==
  +
  +
\(\mathrm{Filog}(z)\) expresses the [[fixed point]] of [[logarithm]] to base \(b\!=\!\exp(z)\).
  +
  +
Another fixed point to the same base can be expressed with
  +
\[\mathrm{Filog}(z^*)^*\]
  +
  +
==Algorithm of evaluation==
  +
[[Filog]] is expressed through the [[Tania function]]:
  +
\[\mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}\]
  +
  +
==Representation of the function==
  +
\(f=\mathrm{Filog}(x+\mathrm{i} y)\) is shown in the \(x,y\) plane with
  +
  +
levels \(u=\Re(f)=\mathrm{cont}\) and
  +
  +
levels \(v=\Im(f)=\mathrm{cont}\); thick lines correspond to the integer values.
  +
  +
The additional thin gridlines \(x\!=\!\exp(-1)\) and \(x\!=\!\pi/2\) are drawn. The first of them goes through the branchpoint \(z=1/\mathrm e\), which is the branch point; the second goes through the point \(z=\pi/2\), where the fixed points are \(\pm \mathrm i\).
  +
  +
==Properties of the function==
  +
  +
\(\mathrm{Filog}(z)\) has two singularities at \(z\!=\!0\) and at \(z\!=\!\exp(-1)\); the cutline is directed to the negative part of the real axis.
  +
  +
Except the cutline, the function is holomorphic. At the real values of the argument \(0\!<\!z\!<\!\exp(-1)\), both at the upper side of the cut and at the lower side of the cut, the function has real values;
  +
in particular, at \(z=\ln\big(\sqrt{2}\big)\), there values are integer
  +
<ref name="sqrt2">
  +
https://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
  +
</ref>:
  +
: \(\mathrm{Filog}(z+\mathrm i o)=2\)
  +
: \(\mathrm{Filog}(z-\mathrm i o)=4\)
  +
  +
Approaching the branchpoint, the jump at the cut vanishes:
  +
\[
  +
\lim_{x\rightarrow 1/\mathrm e} \mathrm{Filog}(x+\mathrm i o)= \lim_{x\rightarrow 1/\mathrm e} \mathrm{Filog}(x-\mathrm i o)= \mathrm e
  +
\]
  +
  +
==Generator of curves==
  +
/* Files [[ado.cin]], [[conto.cin]] and [[filog.cin]] should be loaded to the working directory for the compilation of the [[C++]] code below:*/
  +
<pre>
  +
#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"
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=400,M1=M+1;
  +
int N=401,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("filogbig.eps","w");ado(o,2004,2004);
  +
fprintf(o,"1002 1002 translate\n 100 100 scale\n");
  +
DO(m,M1) X[m]=-10.+.05*(m-.2);
  +
DO(n,200)Y[n]=-10.+.05*n;
  +
Y[200]=-.0001;
  +
Y[201]= .0001;
  +
for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
  +
for(m=-10;m<11;m++){M(m,-10)L(m,10)}
  +
for(n=-10;n<11;n++){M( -10,n)L(10,n)}
  +
fprintf(o,".005 W 0 0 0 RGB S\n");
  +
M(exp(-1.),-1)
  +
L(exp(-1.), 1)
  +
M(M_PI/2.,-1)
  +
L(M_PI/2., 1)
  +
fprintf(o,".003 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+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);
  +
p=Re(c);q=Im(c);
  +
if(p>-15. && p<15. && q>-15. && q<15. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=3.;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,".001 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,".001 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,".001 W 0 0 .9 RGB S\n");
  +
for(m=1;m<14;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".004 W .9 0 0 RGB S\n");
  +
for(m=1;m<14;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".004 W .6 0 .6 RGB S\n");
  +
for(m=-11;m<14;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf filogbig.eps");
  +
system( "open filogbig.pdf"); //for mac
  +
// getchar(); system("killall Preview"); // for mac
  +
// Copyleft 2012 by Dmitrii Kouznetsov
  +
}
  +
</pre>
  +
==Generator of labels==
  +
  +
%For the compilation of the [[Latex]] source below, the curves of the [[complex map]] should be already generated and stored in file fIlog.pdf with the [[C++]] code above.
  +
<pre>
  +
\documentclass[12pt]{article} %<br>
  +
\usepackage{geometry} %<br>
  +
\paperwidth 2074pt %<br>
  +
\paperheight 2060pt %<br>
  +
\topmargin -96pt %<br>
  +
\oddsidemargin -80pt %<br>
  +
\textwidth 2090pt %<br>
  +
\textheight 2066pt %<br>
  +
\usepackage{graphicx} %<br>
  +
\usepackage{rotating} %<br>
  +
\newcommand \rot {\begin{rotate}} %<br>
  +
\newcommand \ero {\end{rotate}} %<br>
  +
\newcommand \rme {\mathrm{e}} %<br>
  +
\newcommand \sx {\scalebox} %<br>
  +
\begin{document} %<br>
  +
\begin{picture}(2018,2040) %<br>
  +
\put(50,40){\includegraphics{filogbig}} %<br>
  +
\put(16,2024){\sx{4.3}{$y$}} %<br>
  +
\put(16,1828){\sx{4.2}{$8$}} %<br>
  +
\put(16,1628){\sx{4.2}{$6$}} %<br>
  +
\put(16,1428){\sx{4.2}{$4$}} %<br>
  +
\put(16,1228){\sx{4.2}{$2$}} %<br>
  +
\put(16,1028){\sx{4.2}{$0$}} %<br>
  +
\put(-11,828){\sx{4}{$-2$}} %<br>
  +
\put(-11,628){\sx{4}{$-4$}} %<br>
  +
\put(-11,428){\sx{4}{$-6$}} %<br>
  +
\put(-11,228){\sx{4}{$-8$}} %<br>
  +
\put(-8,0){\sx{4}{$-10$}} %<br>
  +
\put(204,0){\sx{4}{$-8$}} %<br>
  +
\put(404,0){\sx{4}{$-6$}} %<br>
  +
\put(604,0){\sx{4}{$-4$}} %<br>
  +
\put(804,0){\sx{4}{$-2$}} %<br>
  +
\put(1046,0){\sx{4}{$0$}} %<br>
  +
\put(1246,0){\sx{4}{$2$}} %<br>
  +
\put(1446,0){\sx{4}{$4$}} %<br>
  +
\put(1646,0){\sx{4}{$6$}} %<br>
  +
\put(1846,0){\sx{4}{$8$}} %<br>
  +
\put(2036,0){\sx{4.2}{$x$}} %<br>
  +
%\put(40, 2){\sx{.8}{$1/\rme$}} %<br>
  +
%\put(108, 0){\sx{1}{$1$}} %<br>
  +
%\put(164, 2){\sx{.8}{$\pi/2$}} %<br>
  +
\put(1600,1480){\sx{6}{\rot{55}$u\!=\!0$ \ero} } %<br>
  +
\put(270,1240){\sx{6}{\rot{60}$u\!=\!0.2$ \ero} } %<br>
  +
\put(800,1070){\sx{6}{\rot{55}$u\!=\!0.4$ \ero} } %<br>
  +
\put(90,910){\sx{6}{\rot{16}$u\!=\!0$ \ero} } %<br>
  +
\put(286,470){\sx{6}{\rot{70}$u\!=\!-0.2$ \ero} } %<br>
  +
\put(1686,970){\sx{6}{\rot{-30}$u\!=\!-0.2$ \ero} } %<br>
  +
\put(1686,610){\sx{6}{\rot{26}$v\!=\!0.2$ \ero} } %<br>
  +
\put(1316,210){\sx{6}{\rot{-56}$v\!=\!0$ \ero} } %<br>
  +
\put( 330,444){\sx{6}{\rot{5}$v\!=\!-0.4$ \ero} } %<br>
  +
\put( 700,10){\sx{6}{\rot{56}$v\!=\!-0.2$ \ero} } %<br>
  +
\end{picture} %<br>
  +
\end{document} %<br>
  +
  +
%Copyleft 2012 by Dmitrii Kouznetsov
  +
</pre>
  +
  +
The resulting [[PDF]] file is converted to [[PNG]] with 100 pixels/inch resolution.
  +
  +
==Refewnces==
  +
{{ref}}
  +
  +
{{fer}}
  +
  +
==Keywords==
  +
«[[Complex map]]»,
  +
«[[Fixed point]]»,
  +
«[[Filog]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[Tania function]]»,
  +
«[[Tetration]]»,
  +
  +
«[[Суперфункции]]»,
  +
  +
[[Category:Complex map]]
  +
[[Category:Tania function]]
  +
[[Category:Tetration]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Latex]]
  +
[[Category:Superfunctions]]

Latest revision as of 15:06, 3 January 2026


Filogmap300.png
The zoom-in of the central part of the map

Fig.18.1 from page 247 of book «Superfunctions» [1], 2020.

The same picture appears also as Рис.18.1 at page 254 of the Russian version «Суперфункции» [2], 2014.

The figure shows the Complex map of function Filog.

The zooming of the central part of this map is shown in figure at tight.

Semantics of Filog

\(\mathrm{Filog}(z)\) expresses the fixed point of logarithm to base \(b\!=\!\exp(z)\).

Another fixed point to the same base can be expressed with \[\mathrm{Filog}(z^*)^*\]

Algorithm of evaluation

Filog is expressed through the Tania function: \[\mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}\]

Representation of the function

\(f=\mathrm{Filog}(x+\mathrm{i} y)\) is shown in the \(x,y\) plane with

levels \(u=\Re(f)=\mathrm{cont}\) and

levels \(v=\Im(f)=\mathrm{cont}\); thick lines correspond to the integer values.

The additional thin gridlines \(x\!=\!\exp(-1)\) and \(x\!=\!\pi/2\) are drawn. The first of them goes through the branchpoint \(z=1/\mathrm e\), which is the branch point; the second goes through the point \(z=\pi/2\), where the fixed points are \(\pm \mathrm i\).

Properties of the function

\(\mathrm{Filog}(z)\) has two singularities at \(z\!=\!0\) and at \(z\!=\!\exp(-1)\); the cutline is directed to the negative part of the real axis.

Except the cutline, the function is holomorphic. At the real values of the argument \(0\!<\!z\!<\!\exp(-1)\), both at the upper side of the cut and at the lower side of the cut, the function has real values; in particular, at \(z=\ln\big(\sqrt{2}\big)\), there values are integer [3]:

\(\mathrm{Filog}(z+\mathrm i o)=2\)
\(\mathrm{Filog}(z-\mathrm i o)=4\)

Approaching the branchpoint, the jump at the cut vanishes: \[ \lim_{x\rightarrow 1/\mathrm e} \mathrm{Filog}(x+\mathrm i o)= \lim_{x\rightarrow 1/\mathrm e} \mathrm{Filog}(x-\mathrm i o)= \mathrm e \]

Generator of curves

/* Files ado.cin, conto.cin and filog.cin should be loaded to the working directory for the compilation of the C++ code below:*/ 

 #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"
 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=400,M1=M+1;
 int N=401,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("filogbig.eps","w");ado(o,2004,2004);
 fprintf(o,"1002 1002 translate\n 100 100 scale\n");
 DO(m,M1) X[m]=-10.+.05*(m-.2);
 DO(n,200)Y[n]=-10.+.05*n;
         Y[200]=-.0001;
         Y[201]= .0001;
 for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
 for(m=-10;m<11;m++){M(m,-10)L(m,10)}
 for(n=-10;n<11;n++){M( -10,n)L(10,n)}
 fprintf(o,".005 W 0 0 0 RGB S\n");
 M(exp(-1.),-1)
 L(exp(-1.), 1)
 M(M_PI/2.,-1)
 L(M_PI/2., 1)
 fprintf(o,".003 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+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);
 p=Re(c);q=Im(c);  
 if(p>-15. && p<15. &&  q>-15. && q<15. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
        }}
 fprintf(o,"1 setlinejoin 1 setlinecap\n");  p=3.;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,".001 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,".001 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,".001 W 0 0 .9 RGB S\n");
 for(m=1;m<14;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".004 W .9 0 0 RGB S\n");
 for(m=1;m<14;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 .9 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".004 W .6 0 .6 RGB S\n");
 for(m=-11;m<14;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf filogbig.eps");    
      system(    "open filogbig.pdf"); //for mac
 //    getchar(); system("killall Preview"); // for mac
 // Copyleft 2012 by Dmitrii Kouznetsov
 }

Generator of labels

%For the compilation of the Latex source below, the curves of the complex map should be already generated and stored in file fIlog.pdf with the C++ code above. 

\documentclass[12pt]{article} %<br>
\usepackage{geometry}  %<br>
\paperwidth 2074pt %<br>
\paperheight 2060pt %<br>
\topmargin -96pt %<br>
\oddsidemargin -80pt %<br>
\textwidth 2090pt %<br>
\textheight 2066pt %<br>
\usepackage{graphicx} %<br>
\usepackage{rotating} %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\newcommand \rme {\mathrm{e}} %<br>
\newcommand \sx {\scalebox} %<br>
\begin{document} %<br>
\begin{picture}(2018,2040) %<br>
\put(50,40){\includegraphics{filogbig}} %<br>
\put(16,2024){\sx{4.3}{$y$}} %<br>
\put(16,1828){\sx{4.2}{$8$}} %<br>
\put(16,1628){\sx{4.2}{$6$}} %<br>
\put(16,1428){\sx{4.2}{$4$}} %<br>
\put(16,1228){\sx{4.2}{$2$}} %<br>
\put(16,1028){\sx{4.2}{$0$}} %<br>
\put(-11,828){\sx{4}{$-2$}} %<br>
\put(-11,628){\sx{4}{$-4$}} %<br>
\put(-11,428){\sx{4}{$-6$}} %<br>
\put(-11,228){\sx{4}{$-8$}} %<br>
\put(-8,0){\sx{4}{$-10$}} %<br>
\put(204,0){\sx{4}{$-8$}} %<br>
\put(404,0){\sx{4}{$-6$}} %<br>
\put(604,0){\sx{4}{$-4$}} %<br>
\put(804,0){\sx{4}{$-2$}} %<br>
\put(1046,0){\sx{4}{$0$}} %<br>
\put(1246,0){\sx{4}{$2$}} %<br>
\put(1446,0){\sx{4}{$4$}} %<br>
\put(1646,0){\sx{4}{$6$}} %<br>
\put(1846,0){\sx{4}{$8$}} %<br>
\put(2036,0){\sx{4.2}{$x$}} %<br>
%\put(40, 2){\sx{.8}{$1/\rme$}} %<br>
%\put(108, 0){\sx{1}{$1$}} %<br>
%\put(164, 2){\sx{.8}{$\pi/2$}} %<br>
\put(1600,1480){\sx{6}{\rot{55}$u\!=\!0$ \ero} } %<br>
\put(270,1240){\sx{6}{\rot{60}$u\!=\!0.2$ \ero} } %<br>
\put(800,1070){\sx{6}{\rot{55}$u\!=\!0.4$ \ero} } %<br>
\put(90,910){\sx{6}{\rot{16}$u\!=\!0$ \ero} } %<br>
\put(286,470){\sx{6}{\rot{70}$u\!=\!-0.2$ \ero} } %<br>
\put(1686,970){\sx{6}{\rot{-30}$u\!=\!-0.2$ \ero} } %<br>
\put(1686,610){\sx{6}{\rot{26}$v\!=\!0.2$ \ero} } %<br>
\put(1316,210){\sx{6}{\rot{-56}$v\!=\!0$ \ero} } %<br>
\put( 330,444){\sx{6}{\rot{5}$v\!=\!-0.4$ \ero} } %<br>
\put( 700,10){\sx{6}{\rot{56}$v\!=\!-0.2$ \ero} } %<br>
\end{picture} %<br>
\end{document}   %<br>

 %Copyleft 2012 by Dmitrii Kouznetsov

The resulting PDF file is converted to PNG with 100 pixels/inch resolution.

Refewnces

  1. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
  2. ↑ https://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  3. ↑ https://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

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