ArcLambertW

Fig.1. $$y\!=\!\mathrm{zex}(x)~$$
Fig.2. Complex map, $$u\!+\!\mathrm i v\!=\!\mathrm{zex}(x\!+\!\mathrm i y)$$

ArcLambertW, called for simplicity zex, is elementary function defined with

(1) $$~ ~ ~ \mathrm{ArcLambertW}(z) = \mathrm{zex}(z) = z \exp(z)$$

One of the inverse function of ArcLambertW is called LambertW [1][2] or also ProductLog [3]. In wide ranges of values of $$z$$, the relations

(2) $$~ ~ ~ \mathrm{LambertW}\Big(\mathrm{zex}(z)\Big) = z$$

and

(3) $$~ ~ ~ \mathrm{zex}\Big(\mathrm{LambertW}(z)\Big) = z$$

hold.

For shortness, in expressions, name $$\mathrm{zex}$$ is used instead of $$\mathrm {ArcLambertW}$$.

SuZex and AuZex

ArcLambertW can be treated as a transfer function. Its superfunction is denoted as SuZex. This function satisfies the transfer equation

(4) $$~ ~ ~ \mathrm{zex}\Big( \mathrm{SuZex}(z) \Big)=\mathrm{SuZex}(z\!+\!1)$$

The complex map of SuZex is available in the category zex below.

The Abel function of ArcLambertW is named AuZex; $$\mathrm{AuZex}=\mathrm{SuZex}^{-1}$$. This function satisfies the Abel equation

(5) $$~ ~ ~ \mathrm{AuZex}\Big( \mathrm{zex}(z) \Big)=\mathrm{AuZex}(z)+1$$

Iterations of zex

Fig.3. Iterates $$~y\!=\!\mathrm{zex}^n(x)~$$ for various values of $$n$$

With superfunction of zex, called SuZex, and the Abel function, called AuZex, the iterations of zex can be expressed as follows:

(8) $$~ ~ ~\mathrm{zex}^n(z)=\mathrm{SuZex}\Big( n+ \mathrm{AuZex}(z)\Big)$$

The number $$n$$ of iteration hax no need to be integer. For several real values of $$n$$, the $$n$$th iteration of zex by (8) is plotted in figure at right,

(9) $$~ ~ ~y=\mathrm{zex}^c(x)$$

At $$n\!=\!2$$, this gives $$y\!=\!\mathrm{zex}^2(x)\!=\!\mathrm{zex}\Big( \mathrm{zex}(x)\Big)$$
at $$n\!=\!1$$, this gives $$y\!=\!\mathrm{zex}(x)\!=\! x \exp(x)$$
at $$n\!=\!0$$, this gives identity function $$y\!=\!\mathrm{zex}^0(x)\!=\! x$$
at $$n\!=\!-1$$, this gives function $$y\!=\!\mathrm{zex}^{-1}(x)\!=\!\mathrm{LambertW}(x) ~$$, and so on.

Figure 3 looks pretty similar to the plots of iterates of other functions with fast growth along the real axis.

Expression (8) can be used for evaluation of funtion LambertW; however, such an evaluation is neither faster, nor preciser that that by the direct implementation of LambertW suggested in LambertW.cin.

The implementation of SuZex and AuZex is pretty similar to that of tetration to base $$\exp^2(-1)$$ [4].