Notebook Principal Branch not working yet

Question

This notebook again is useful to see in a 3D plot of the complex log function :

• the principal branch
• the branches

For this purpose, I downloaded the notebook from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/PrincipalBranch.nb.

I also downloaded the Utilities`Typesetting` package of Eric Weisstein First, it goes wrong to see the "branches" in a 3D plot and then it goes wrong with the "principal(branch)" 3D plot. I start with the branches calculation.

<< Utilities`Typesetting`

branches = 
 Show[Graphics3D[
                 ParametricPlot3D[
                                 Evaluate[
 Table[{r Cos[θ], r Sin[θ], 
 Im[Log[r Exp[I θ]]] + 2 Pi k}, {k, -1, 1}]], {r, .01, 
 1}, {θ, 0, 2 π},
 DisplayFunction -> Identity,
 PlotPoints -> {4, 25}][[1]] /. 
 Polygon[x_] :> 
 If[Max[z = Last /@ x] - Min[z] > .5, 
 Polygon[Join[# - {0, 0, 2 Pi} & /@ Take[x, 2], Take[x, -2]]], 
 Polygon[x]]],
 Axes -> Automatic,
 AxesLabel -> TraditionalForm /@ {x, y, HoldForm[Im[Log[z]]]},
 AxesEdge -> {{-1, -1}, {1, -1}, {1, 1}},
 BoxRatios -> {1, 1, 1},
 PlotLabel -> 
 StyleForm["multiple branches", FontSlant -> "Italic", 
 FontFamily -> "Times"],
 PlotRange -> {-10, 10}]

Note: don't know yet how to write the code here in a programmer style

Answer 1

That's old, unnecessarilly-complex code. Start with: $$ \begin{align} \log(z)&=\ln|z|+i\arg(z) \\ &=\ln|z|+i\left[\text{Arg}(z)+2 k\pi\right];\quad -\pi\lt\text{Arg}(z)\leq \pi \end{align} $$ $\ln|z|$ is the single-valued real component but $\arg(z)$ is the multivalued imaginary component. $k=0$ is the principal branch of the imgainary component. Simpler code to plot branching $k=0,1,2$:

f[z_, k_] = Log[Abs[z]] + I (Arg[z] + 2 k Pi)
realPlot = 
  ParametricPlot3D[{Re[z], Im[z], Re[f[z, 0]]} /. z -> r Exp[I t], {r,
     0.001, 1}, {t, -Pi, Pi}, PlotLabel -> Style["Real Sheet", 16]];
colors = {Red, Blue, Green};
imagPlotList = Table[
   ParametricPlot3D[{Re[z], Im[z], Im[f[z, i - 1]]} /. 
     z -> r Exp[I t], {r, 0.001, 1}, {t, -Pi, Pi}, 
    PlotStyle -> colors[[i]]],
   {i, 1, 3}
   ];
imagPlot = 
  Show[imagPlotList, PlotRange -> All, BoxRatios -> {1, 1, 1}, 
   PlotLabel -> Style["Imag sheets", 16]];
GraphicsRow[{realPlot, imagPlot}]

Answer 2

Eric's code includes a replacement /. Polygon[x_] :> … because there's no Exclusions option in v5.2. Nowadays the code can be simplified to

ParametricPlot3D[
 Evaluate@Table[{r Cos[θ], r Sin[θ], Im[Log[r Exp[I θ]]] + 2 Pi k}, {k, -1, 1}], 
                {r, .01, 1}, {θ, 0, 2 π}, BoxRatios -> {1, 1, 1}]

ParametricPlot3D[{r Cos[θ], r Sin[θ], Im[Log[r Exp[I θ]]]}, 
                 {r, .01, 1}, {θ, 0, 2 π},
                 BoxRatios -> {1, 1, 1}, 
                 PlotRange -> {Automatic, Automatic, {-10, 10}}]

For comparison, The followings are outputs in v5.2:

Just for fun, the following is a "quick" fix based on the logic of Eric's code, notice I've forced Exclusions to be None:

branches = Show[
  Graphics3D[Normal@
      ParametricPlot3D[
        Evaluate[Table[{r Cos[θ], r Sin[θ], 
           Im[Log[r Exp[I θ]]] + 2 Pi k}, {k, -1, 1}]], {r, .01, 1}, {θ, 0,
          2 π}, PlotPoints -> {4, 25}, Exclusions -> None][[1]] /. 
     Line[pts_] :> Line[Split[pts, Abs@Last[#2 - #1] < Pi &]] /. 
    Polygon[x_, opt__] :> 
     If[Max[z = Last /@ x] - Min[z] > 
       Pi, {Polygon[Join[# - {0, 0, 2 Pi} & /@ Take[x, 2], Take[x, -2]], opt]}, {Polygon[
        x, opt]}]], Axes -> Automatic, 
  AxesLabel -> TraditionalForm /@ {x, y, HoldForm[Im[Log[z]]]}, 
  AxesEdge -> {{-1, -1}, {1, -1}, {1, 1}}, BoxRatios -> {1, 1, 1}, 
  PlotLabel -> 
   StyleForm["multiple branches", FontSlant -> "Italic", FontFamily -> "Times"], 
  PlotRange -> {-10, 10}]

A new replacement /. Line[pts_] :> Line[Split[pts, Abs@Last[#2 - #1] < Pi &]] is added because Mesh option seems to be working in a different manner nowadays. (Not 100% sure what's the difference though. )