File:Filogbigmap100.png

The zoom-in of the central part of the map

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 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 ^[1]:

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

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

Approaching 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 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.

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

Rwfwewnces

Keywords

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