0
$\begingroup$

I have expressions $\log\frac{z}{z-1}$ and $\frac{1}{z \left( 1 + W_0\left(-\frac{1}{e z}\right) \right)} $ visualized below. There's a branch cut along the Im[z]=0 plane, what is an efficient way to visualize their branch cuts in a 2D plot?

enter image description here

plotStieltjes[expr_] := ParametricPlot3D[{x, y^3/(1/2)^2, Im@expr /. z -> (x + I y^3/(1/2)^2)}, {x, -1/2, 3/2}, {y, -1/2, 1/2}, PlotRange -> {Automatic, Automatic, {-5, 5}}, PlotPoints -> 25, MaxRecursion -> 4, ColorFunction -> (ColorData["TemperatureMap"][ Rescale[#3, {4, -4}, {0, 1}]] &), ColorFunctionScaling -> False, ExclusionsStyle -> {None, Directive[Red, Thick]}, MeshFunctions -> {#3 &}, Mesh -> {Range[-3, 3, .5]}, MeshStyle -> {{Opacity[0.5], Darker[Blue]}, {Opacity[0.3], Gray}}, PlotStyle -> Directive[Specularity[White, 10], Opacity[0.95]], Exclusions -> {{Im[z] == 0} /. z -> x + I y}, Lighting -> {{"Ambient", GrayLevel[0.3]}, {"Directional", White, ImageScaled[{0, 0, 2}]}, {"Directional", White, ImageScaled[{1, 1, 1}]}}, AxesLabel -> {Style["Re[z]", FontSize -> 14, Bold], Style["Im[z]", FontSize -> 14, Bold], Style["Im[f(z)]", FontSize -> 14, Bold]}, BoxRatios -> {1, 1, 0.6}, ViewPoint -> {-2.5, 2.5, 1.3}, ViewVertical -> {0, 0, 1}, PlotTheme -> "Detailed", Background -> GrayLevel[0.95], ImageSize -> Large, AxesStyle -> Directive[Black, Thickness[0.002]], Boxed -> False, FaceGrids -> None, Ticks -> {{0, 1}, None, {-Pi, 0, Pi}}, BaseStyle -> {FontFamily -> "Helvetica"}] plotStieltjes[Log[z/(z - 1)]] plotStieltjes[1/(z (1 + ProductLog[-(1/(E z))]))]
Improve this question
$\endgroup$
10
  • 2
    $\begingroup$ How is this question different from your previous one? mathematica.stackexchange.com/questions/313994/… $\endgroup$ Commented Jun 10 at 21:57
  • $\begingroup$ @azerbajdzan "visualize their branch cuts in a 2D plot" $\endgroup$ Commented Jun 10 at 22:23
  • $\begingroup$ So you set z->x+0 I and plot it for {x,0,1}, no? $\endgroup$ Commented Jun 10 at 22:25
  • 2
    $\begingroup$ This question is similar to: Plotting branch-cuts along Re[z] line. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. $\endgroup$ Commented Jun 10 at 22:52
  • $\begingroup$ @DavidG.Stork this question is specific to a 2D plot of the branch cut $\endgroup$ Commented Jun 10 at 23:11

1 Answer 1

Reset to default
2
$\begingroup$

Use Limit[] to compute the boundary curves $w^*=b(x)$ of $w=f(z)$ and plot $(x,\text{Im}(w^*))$ parametrically.

FunctionSingularities[ 1/(z (1 + ProductLog[-(1/(E z))])), z, Complexes] // Reduce (* (0 < Re[z] <= 1 && Im[z] == 0) || z == 0 || z == 1 *) posedge = Limit[1/(z (1 + ProductLog[-(1/(E z))])) /. z -> x + I y, y -> 0, Direction -> "FromAbove", Assumptions -> 0 < x < 1]; line1 = Normal@ParametricPlot[{x, Im@posedge}, {x, 0, 1}]; negedge = Limit[1/(z (1 + ProductLog[-(1/(E z))])) /. z -> x + I y, y -> 0, Direction -> "FromBelow", Assumptions -> 0 < x < 1]; line2 = Normal@ ParametricPlot[{x, Im@negedge}, {x, 0, 1}, ColorFunction -> "Rainbow"]; Graphics[ {AbsoluteThickness[2], Red, Line[ Join[ First@Cases[line1, Line[p_] :> p, Infinity], First@Cases[line2, Line[p_] :> Reverse@p, Infinity] ] // Append[#, First@#] & ] }, PlotRange -> {{-0.5, 1.5}, {-4, 4}}, Axes -> True, AspectRatio -> 0.7]

plot of the branch cut

Don't join the lines if you don't want the "asymptotes."

Improve this answer
$\endgroup$
2
  • $\begingroup$ This does not work on version 13.0.1. The output for me is just the lower half of the branch cut. $\endgroup$ Commented Jun 12 at 10:44
  • $\begingroup$ @azerbajdzan Most likely there is a problem computing negedge, which is the same as posedge except for the branch ProductLog[-1,...] instead of ProductLog[...]. Maybe it's a bug in V13, because it seems a simple limit. I don't have V13, so I can't do anything but guess. $\endgroup$ Commented Jun 12 at 10:55

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.