File:Filogmap300.png

Complex map of function Filog.

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:

$\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z}$

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 gridlines $x\!=\!\exp(-1)$ and $x\!=\!\pi/2$ are drawn. The first of them goes through the branchpoint $z=1/\mathrm e$; 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 ^[1]:

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

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

At the branchpoint, the jump at the cut vanishes:

$ \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$

Generator of curves

// Files ado.cin, conto.cin and filog.cin should be loaded to the working directory for the compillation of 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++)
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=200,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("filog.eps","w");ado(o,202,202);
fprintf(o,"1 101 translate\n 100 100 scale\n");
DO(m,M1) X[m]=0.+.01*(m-.2);
DO(n,200)Y[n]=-1.+.005*n;
        Y[200]=-.0001;
        Y[201]= .0001;
for(n=202;n<N1;n++) Y[n]=-1.+.005*(n-1.);
for(m=0;m<3;m++){M(m,-1)L(m,1)}
for(n=-1;n<2;n++){M(  0,n)L(2,n)}
fprintf(o,".004 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,".002 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 filog.eps");    
      system(    "open filog.pdf"); //for mac
//    getchar(); system("killall Preview"); // for mac
}
// Copyleft 2012 by Dmitrii Kouznetsov

Generator of labels

For the compillation 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 215pt %<br> \paperheight 216pt %<br> \topmargin -102pt %<br> \oddsidemargin -88pt %<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}(208,208) %<br> \put(10,10){\includegraphics{filog}} %<br> \put(4,207){$1$} %<br> \put(4,107){$0$} %<br> \put(-3, 8){$-\!1$} %<br> \put(8, 0){$0$} %<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(206, 2){\sx{.9}{$x$}} %<br> \put(85,43){\sx{.8}{\rot{64}$v\!=\!2$ \ero} } %<br> \put(150,49){\sx{.6}{\rot{84}$v\!=\!1.2$ \ero} } %<br> \put(181.5,43){\sx{.8}{\rot{80}$v\!=\!1$ \ero} } %<br> \put(183,161){\sx{.6}{\rot{-40}$v\!=\!0.8$ \ero} } %<br> \put(150,143){\sx{.6}{\rot{60}$u\!=\!0.2$ \ero} } %<br> \put(183,118){\sx{.8}{\rot{35}$u\!=\!0$ \ero} } %<br> \put(184,74){\sx{.6}{\rot{5}$u\!=\!-0.2$ \ero} } %<br> \end{picture} %<br> \end{document} %<br> %Copyleft 2012 by Dmitrii Kouznetsov

Rwfwewnces

Keywords

Fixed point, Filog, Tania function, Tetration, Complex map