File:FilogbigT.jpg

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

Another fixed point to the same base can be expressed with

${\displaystyle \mathrm {Filog} (z^{*})^{*}}$

Algorithm of evaluation

Filog is expressed through the Tania function:

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

Representation of the function

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

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

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

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

Properties of the function

${\displaystyle \mathrm {Filog} (z)}$ has two singularities at ${\displaystyle z\!=\!0}$ and at ${\displaystyle z\!=\!\exp(-1)}$; 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 ${\displaystyle 0\!<\!z\!<\!\exp(-1)}$, both at the upper side of the cut and at the lower side of the cut, the function has real values; in particular, at ${\displaystyle z=\ln {\big (}{\sqrt {2}}{\big )}}$, there values are integer ^[1]:

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

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

Approaching the branchpoint, the jump at the cut vanishes:

${\displaystyle \displaystyle \lim _{x\rightarrow 1/\mathrm {e} }\mathrm {Filog} (x+\mathrm {i} o)=\lim _{x\rightarrow 1/\mathrm {e} }\mathrm {Filog} (x-\mathrm {i} o)=\mathrm {e} }$

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;}

{{Image|FilogbigT.jpg|right|350px|Add image caption here.}}