File:FilogbigT.jpg: Difference between revisions

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imported>Dmitrii Kouznetsov
(uploaded a new version of "File:FilogbigT.jpg": make gridlines thicker)
imported>Dmitrii Kouznetsov
(remove the gerenator, it is not shown well)
Line 9: Line 9:
<math>\mathrm{Filog}(z^*)^*</math>
<math>\mathrm{Filog}(z^*)^*</math>


==Algorithm of evaluation==
[[Filog]] is expressed through the [[Tania function]], <math>\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}</math>
[[Filog]] is expressed through the [[Tania function]]:
: <math>\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}</math>
 
==Representation of the function==


<math>f=\mathrm{Filog}(x+\mathrm{i} y)</math> is shown in the <math>x,y</math> plane with  
<math>f=\mathrm{Filog}(x+\mathrm{i} y)</math> is shown in the <math>x,y</math> plane with  
Line 23: Line 19:
The additional thin gridlines <math>x\!=\!\exp(-1)</math> and <math>x\!=\!\pi/2</math> are drawn. The first of them goes through the branchpoint <math>z=1/\mathrm e</math>, which is the branch point; the second goes through the point  
The additional thin gridlines <math>x\!=\!\exp(-1)</math> and <math>x\!=\!\pi/2</math> are drawn. The first of them goes through the branchpoint <math>z=1/\mathrm e</math>, which is the branch point; the second goes through the point  
<math>z=\pi/2</math>, where the fixed points are </math>\pm \mathrm i</math>.
<math>z=\pi/2</math>, where the fixed points are </math>\pm \mathrm i</math>.
==Properties of the function==
<math>\mathrm{Filog}(z)</math> has two singularities at
<math>z\!=\!0</math> and at
<math>z\!=\!\exp(-1)</math>; 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 <math>0\!<\!z\!<\!\exp(-1)</math>, both at the upper side of the cut and at the lower side of the cut, the function has real values; in particular, at
<math>z=\ln\big(\sqrt{2}\big)</math>, there values are integer
<ref name="sqrt2">
http://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>:
: <math>\mathrm{Filog}(z+\mathrm i o)=2</math>
: <math>\mathrm{Filog}(z-\mathrm i o)=4</math>
Approaching the branchpoint, the jump at the cut vanishes:
: <math> \displaystyle \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</math>
==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
filogbig.pdf with the [[C++]] code above.
<nowiki>
\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}{</math>y</math>}} %<br>
\put(16,1828){\sx{4.2}{</math>8</math>}} %<br>
\put(16,1628){\sx{4.2}{</math>6</math>}} %<br>
\put(16,1428){\sx{4.2}{</math>4</math>}} %<br>
\put(16,1228){\sx{4.2}{</math>2</math>}} %<br>
\put(16,1028){\sx{4.2}{</math>0</math>}} %<br>
\put(-11,828){\sx{4}{</math>-2</math>}} %<br>
\put(-11,628){\sx{4}{</math>-4</math>}} %<br>
\put(-11,428){\sx{4}{</math>-6</math>}} %<br>
\put(-11,228){\sx{4}{</math>-8</math>}} %<br>
\put(-8,0){\sx{4}{</math>-10</math>}} %<br>
\put(204,0){\sx{4}{</math>-8</math>}} %<br>
\put(404,0){\sx{4}{</math>-6</math>}} %<br>
\put(604,0){\sx{4}{</math>-4</math>}} %<br>
\put(804,0){\sx{4}{</math>-2</math>}} %<br>
\put(1046,0){\sx{4}{</math>0</math>}} %<br>
\put(1246,0){\sx{4}{</math>2</math>}} %<br>
\put(1446,0){\sx{4}{</math>4</math>}} %<br>
\put(1646,0){\sx{4}{</math>6</math>}} %<br>
\put(1846,0){\sx{4}{</math>8</math>}} %<br>
\put(2036,0){\sx{4.2}{</math>x</math>}} %<br>
%\put(40, 2){\sx{.8}{</math>1/\rme</math>}} %<br>
%\put(108, 0){\sx{1}{</math>1</math>}} %<br>
%\put(164, 2){\sx{.8}{</math>\pi/2</math>}} %<br>
\put(1600,1480){\sx{6}{\rot{55}</math>u\!=\!0</math> \ero} } %<br>
\put(270,1240){\sx{6}{\rot{60}</math>u\!=\!0.2</math> \ero} } %<br>
\put(800,1070){\sx{6}{\rot{55}</math>u\!=\!0.4</math> \ero} } %<br>
\put(90,910){\sx{6}{\rot{16}</math>u\!=\!0</math> \ero} } %<br>
\put(286,470){\sx{6}{\rot{70}</math>u\!=\!-0.2</math> \ero} } %<br>
\put(1686,970){\sx{6}{\rot{-30}</math>u\!=\!-0.2</math> \ero} } %<br>
\put(1686,610){\sx{6}{\rot{26}</math>v\!=\!0.2</math> \ero} } %<br>
\put(1316,210){\sx{6}{\rot{-56}</math>v\!=\!0</math> \ero} } %<br>
\put( 330,444){\sx{6}{\rot{5}</math>v\!=\!-0.4</math> \ero} } %<br>
\put( 700,10){\sx{6}{\rot{56}</math>v\!=\!-0.2</math> \ero} } %<br>
\end{picture} %<br>
\end{document} %<br>
%Copyleft 2012 by Dmitrii Kouznetsov
</nowiki>
The resulting [[PDF]] file is converted to [[PNG]] with 100 pixels/inch resolution.
==Rwfwewnces==
<references/>
==Keywords==
[[Fixed point]],
[[Filog]],
[[Tania function]],
[[Tetration]],
[[Complex map]]


|author      = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
|author      = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]

Revision as of 22:15, 7 March 2012

Summary

Title / Description


Complex map of function Filog.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Filog}(z)} expresses the fixed point of logarithm to base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\!=\!\exp(z)} .

Another fixed point to the same base can be expressed with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Filog}(z^*)^*}

Filog is expressed through the Tania function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\mathrm{Filog}(x+\mathrm{i} y)} is shown in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} plane with

levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\Re(f)=\mathrm{cont}} and

levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=\Im(f)=\mathrm{cont}} ; thick lines correspond to the integer values.

The additional thin gridlines and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\!=\!\pi/2} are drawn. The first of them goes through the branchpoint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=1/\mathrm e} , which is the branch point; the second goes through the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=\pi/2} , where the fixed points are </math>\pm \mathrm i</math>.

Citizendium author
& Copyright holder


Copyright © Dmitrii Kouznetsov.
See below for licence/re-use information.
Date created


2012.03.08
Country of first publication


Japan
Notes


I tried to save it as http://en.citizendium.org/wiki/File:FilogmapT.png

but it does not load as it is expected..

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File:FilogmapT.png and http://tori.ils.uec.ac.jp/TORI/index.php/File:Filogbigmap100.png
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