Editing Lambert W function (section)

== Elementary properties, branches and range ==
[[File:Lambert W Range.pdf|thumb|left|upright=1.1|The range of the {{mvar|W}} function, showing all branches. The black curves (including the real axis) form the image of the real axis, the orange curves are the image of the imaginary axis. The purple curve and circle are the image of a small circle around the point {{math|1=''z'' = 0}}; the red curves are the image of a small circle around the point {{math|1=''z'' = −1/e}}.]]

[[File:Imaginary_part_of_the_Lambert_W(n,x%2Bi_y)_for_various_branches_(n).jpg|thumb|right|upright=1.3|Plot of the imaginary part of {{math|''W''{{sub|''n''}}(''x'' + ''iy'')}} for branches {{math|1=''n'' = −2, −1, 0, 1, 2}}. The plot is similar to that of the multivalued [[complex logarithm]] function except that the spacing between sheets is not constant and the connection of the principal sheet is different]]

There are countably many branches of the {{mvar|W}} function, denoted by {{math|''W<sub>k</sub>''(''z'')}}, for integer {{mvar|k}}; {{math|''W''<sub>0</sub>(''z'')}} being the main (or principal) branch. {{math|''W''<sub>0</sub>(''z'')}} is defined for all complex numbers ''z'' while {{math|''W<sub>k</sub>''(''z'')}} with {{math|''k'' ≠ 0}} is defined for all non-zero ''z''. With {{math|1=''W''<sub>0</sub>(0) = 0}} and {{math|1={{underset|''z''→0|lim}} ''W''<sub>''k''</sub>(''z'') = −∞}} for all {{math|''k'' ≠ 0}}.

The branch point for the principal branch is at {{math|1=''z'' = −{{sfrac|1|''e''}}}}, with a branch cut that extends to {{math|−∞}} along the negative real axis. This branch cut separates the principal branch from the two branches {{math|''W''<sub>−1</sub>}} and {{math|''W''<sub>1</sub>}}. In all branches {{math|''W<sub>k</sub>''}} with {{math|''k'' ≠ 0}}, there is a branch point at {{math|1=''z'' = 0}} and a branch cut along the entire negative real axis.

The functions {{math|''W<sub>k</sub>''(''z''), ''k'' &isin; '''Z'''}} are all [[Injective function|injective]] and their ranges are disjoint. The range of the entire multivalued function {{mvar|W}} is the complex plane. The image of the real axis is the union of the real axis and the [[quadratrix of Hippias]], the parametric curve {{math|1=''w'' = −''t'' cot ''t'' + ''it''}}.
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=== 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}}.
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