File:Lambert W Range.pdf

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Description	The range of the Lambert W function, showing all branches. The orange curves are images of either the positive or the negative imaginary axis. The black curves are images of the positive or negative real axis (except for the one that intersects −1, which is the image of part of the negative real axis). The purple curve and circle is the image of a small circle around the branch point z=0; the red curves are the images of a small circle around the point z=-1/e. The range of W₀ is inside the C-shaped black curve. The range of each of the other branches is a band between two black curves that represent points on the negative real axis (a black curve representing the positive real axis runs through the middle of each such band).
Source	Created in mathematica 9.0
Date	2017-10-06
Author	Sławomir Biały
Permission (Reusing this file)	See below.

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<syntaxhighlight lang="haskell"> ComplexPlot[f_, opts__] := ParametricPlot[{Re[f], Im[f]}, opts] branch[k_] := {ComplexPlot[

  ProductLog[k, -1/E + E^(I t)/10], {t, -\[Pi], \[Pi]}, 
  PlotStyle -> Red], 
 ComplexPlot[ProductLog[k, E^(I t)/5], {t, -\[Pi], \[Pi]}, 
  PlotStyle -> Purple], 
 ComplexPlot[ProductLog[k, I E^t], {t, -10, 2000}, 
  PlotStyle -> Orange, PlotRange -> All, PlotPoints -> 1000],
 ComplexPlot[ProductLog[k, I E^t], {t, -2000, -10}, 
  PlotStyle -> Orange, PlotRange -> All, PlotPoints -> 1000],
 ComplexPlot[ProductLog[k, -I E^t], {t, -10, 2000}, 
  PlotStyle -> Orange, PlotRange -> All, PlotPoints -> 1000],
 ComplexPlot[ProductLog[k, -I E^t], {t, -2000, -10}, 
  PlotStyle -> Orange, PlotRange -> All, PlotPoints -> 1000]}

Show[ParametricPlot[{-t Cot[t], t}, {t, -5 \[Pi], 5 \[Pi]},

 PlotStyle -> {Thickness[0.002], Black}, 
 Exclusions -> Range[-4, 4, 1]*\[Pi]], branch[0], branch[1], 
branch[-1], branch[2], branch[-2], branch[3], branch[-3], 
PlotRange -> {{-16, 16}, {-16, 16}}]

</syntaxhighlight>

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Lambert W function