structural:summarization, information:classification, linguistic:paraphrasing
Inverse

The range plot above also delineates the regions in the complex plane where the simple inverse relationship W (n, z e z) = z {\displaystyle W(n,ze^{z})=z} is true. f = z e z {\displaystyle f=ze^{z}} implies that there exists an n {\displaystyle n} such that z = W (n, f) = W (n, z e z) {\displaystyle z=W(n,f)=W(n,ze^{z})}, where n {\displaystyle n} depends upon the value of z {\displaystyle z}. The value of the integer n {\displaystyle n} changes abruptly when z e z {\displaystyle ze^{z}} is at the branch cut of W (n, z e z) {\displaystyle W(n,ze^{z})}, which means that z e z {\displaystyle ze^{z}} ≤ 0, except for n = 0 {\displaystyle n=0} where it is z e z {\displaystyle ze^{z}} ≤ −1/ e {\displaystyle e}.

Defining z = x + i y {\displaystyle z=x+iy}, where x {\displaystyle x} and y {\displaystyle y} are real, and expressing e z {\displaystyle e^{z}} in polar coordinates, it is seen that

For n ≠ 0 {\displaystyle n\neq 0}, the branch cut for W (n, z e z) {\displaystyle W(n,ze^{z})} is the non-positive real axis, so that

and

For n = 0 {\displaystyle n=0}, the branch cut for W {\displaystyle W} is the real axis with − ∞ < z ≤ − 1 / e {\displaystyle -\infty <z\leq -1/e}, so that the inequality becomes

Inside the regions bounded by the above, there are no discontinuous changes in W (n, z e z) {\displaystyle W(n,ze^{z})}, and those regions specify where the W {\displaystyle W} function is simply invertible, i.e. W (n, z e z) = z {\displaystyle W(n,ze^{z})=z}.