Fixed point of logarithm

Complex map of function $f=\mathrm {Filog} (x+\mathrm {i} y)$

Zoom-in from the central part of the previous figure

Fixed points of logarithm to base $b$ are solutions $L$ of equation

$\!\!\!\!\!(1)~~~\displaystyle L=\log _{b}(L)$

The special name Filog (Fixed point of logarithm) is suggested for the function that expresses one of these solutions $L_{1}$ through the logarithm of the base $b$ ;

$\!\!\!\!\!(2)~~~\displaystyle L_{1}=\mathrm {Filog} (a)={\frac {\mathrm {Tania} \!{\big (}\ln(a)-1-\mathrm {i} {\big )}}{-a}}$

where $a=\ln(b)$ and $\mathrm {Tania}$ is Tania function, id est, solution of the equation

$\!\!\!\!\!(3)~~~\displaystyle \mathrm {Tania} '(z)={\frac {\mathrm {Tania} (z)}{1+\mathrm {Tania} '(z)}}$

with

$\!\!\!\!\!(4)~~~\displaystyle \mathrm {Tania} (0)=1{\frac {\mathrm {Tania} (z)}{1+\mathrm {Tania} '(z)}}$

the prime denotes the derivative.

Then, another fixed point $L_{2}$ can be expressed as follows:

$\!\!\!\!\!(5)~~~\displaystyle L_{2}=\mathrm {Filog} (a^{*})^{*}$

where asterisk (*) denotes the complex conjugation.

The complex map of function $\mathrm {Filog}$ is shown in the top figure at right. $f=\mathrm {Filog} (x+\mathrm {i} y)$ is shown in the $x,y$ plane with levels $u=\Re (f)={\rm {const}}$ and levels $v=\Im (f)={\rm {const}}$. The next figure shows the zoom-in of the central part.

Relation to the LambertW function

While the real part of the base $b$ is positive, the fixed point of logarithm can be expressed also through the LambertW function.

$\!\!\!\!\!(6)~~~\displaystyle L_{1}={\frac {\mathrm {LambertW} (-a)}{-a}}={\frac {\mathrm {LambertW} (-\ln(b))}{-\ln(b)}}$

Both fixed points in the whole complex plane of values of base may be neecessary; then, the representation (3) through the Tania function is more convenient than that through the LambertW; the efficient algorithm for evaluation of the Tania function is available .

Application of the fixed points of logarithm

The efficient evaluation of the fixed points of logarithm is important for the construction and evaluation of the tetration to the complex base. The tetration approaches values $L_{1}$ or $L_{2}$ as the imaginary part of the argument becomes large.