Editing Lambert W function (section)

=== Inverse ===
[[File:InvertW.jpg|thumb|upright=1.3|Regions of the complex plane for which {{math|1=''W''(''n'', ''ze''<sup>''z''</sup>) = ''z''}}, where {{math|1= ''z'' = ''x'' + ''iy''}}. The darker boundaries of a particular region are included in the lighter region of the same color. The point at {{math|{{brace|−1,&nbsp;0}}}} is included in both the {{math|1=''n'' = −1}} (blue) region and the {{math|1=''n'' = 0}} (gray) region. Horizontal grid lines are in multiples of {{math|''π''}}.]]

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

Defining {{tmath|1=z = x + iy}}, where {{tmath|1=x}} and {{tmath|1=y}} are real, and expressing {{tmath|1=e^z}} in polar coordinates, it is seen that
: <math>
\begin{align}
 ze^z &= (x + iy) e^{x} (\cos y + i \sin y) \\
       &= e^{x} (x \cos y - y \sin y) + i e^{x} (x \sin y + y \cos y) \\
\end{align}
</math>

For <math>n \neq 0</math>, the branch cut for {{tmath|1=W(n, ze^z)}} is the non-positive real axis, so that
: <math>x \sin y + y \cos y = 0 \Rightarrow x = -y/\tan(y),</math>
and
: <math>(x \cos y - y \sin y) e^x \leq 0.</math>
For <math>n = 0</math>, the branch cut for {{tmath|1=W[n,z e^z]}} is the real axis with <math>-\infty < z \leq -1/e</math>, so that the inequality becomes
: <math>(x \cos y - y \sin y) e^x \leq -1/e.</math>

Inside the regions bounded by the above, there are no discontinuous changes in {{tmath|1=W(n, ze^z)}}, and those regions specify where the {{tmath|1=W}} function is simply invertible, i.e. {{tmath|1=W(n, ze^z) = z}}.
{{Clear}}