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

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:

using namespace std; typedef complex z_type; 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-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
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"
 * 5) include "filog.cin"

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} % \usepackage{geometry} % \paperwidth 215pt % \paperheight 216pt % \topmargin -102pt % \oddsidemargin -88pt % \usepackage{graphicx} % \usepackage{rotating} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \rme {\mathrm{e}} % \newcommand \sx {\scalebox} % \begin{document} % \begin{picture}(208,208) % \put(10,10){\includegraphics{filog}} % \put(4,207){$1$} % \put(4,107){$0$} % \put(-3, 8){$-\!1$} % \put(8, 0){$0$} % \put(40, 2){\sx{.8}{$1/\rme$}} % \put(108, 0){\sx{1}{$1$}} % \put(164, 2){\sx{.8}{$\pi/2$}} % \put(206, 2){\sx{.9}{$x$}} % \put(85,43){\sx{.8}{\rot{64}$v\!=\!2$ \ero} } % \put(150,49){\sx{.6}{\rot{84}$v\!=\!1.2$ \ero} } % \put(181.5,43){\sx{.8}{\rot{80}$v\!=\!1$ \ero} } % \put(183,161){\sx{.6}{\rot{-40}$v\!=\!0.8$ \ero} } % \put(150,143){\sx{.6}{\rot{60}$u\!=\!0.2$ \ero} } % \put(183,118){\sx{.8}{\rot{35}$u\!=\!0$ \ero} } % \put(184,74){\sx{.6}{\rot{5}$u\!=\!-0.2$ \ero} } % \end{picture} % \end{document}  % %Copyleft 2012 by Dmitrii Kouznetsov

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