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

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

Filog is expressed through the Tania function, ${\displaystyle \displaystyle \mathrm {Filog} (z)={\frac {\mathrm {Tania} \!{\big (}\ln(z)-1-\mathrm {i} {\big )}}{-z}}}$

${\displaystyle f=\mathrm {Filog} (x+\mathrm {i} y)}$ 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>.

but it does not load as it is expected..

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