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I want to draw an illustration in the book Visual Complex Analysis(ISBN: 9780198534464):

enter image description here

ComplexPlot3D[Abs[Sin[z]], {z, -4 - 4 I, 4 + 4 I}, 
 RegionFunction -> 
  Function[{z, 
    f}, (π/2 <= Arg[z] < π) || (-π < Arg[z] <= 0)], 
 Filling -> Bottom, FillingStyle -> Directive[Opacity[0.7], Gray], 
 Mesh -> {100, 50}, MeshFunctions -> {Log[Abs[#2]] &, Log[Abs[#1]] &},
  PlotRange -> {0, Pi}, ColorFunction -> "CyclicLogAbsArg"]

But the vertical grid in this picture is obviously different from that in the textbook:

enter image description here

How to set the parameters of function MeshFunctions correctly to get the same grid as in the textbook?

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According to Markushevich,Theory of Functions of a Complex Variable,1965, P149,the MeshFunction should be Arg[Sin[z]],

ComplexPlot3D[Abs[Sin[z]], {z, -4 - 4 I, 4 + 4 I}, 
 RegionFunction -> 
  Function[{z, 
    f}, (π/2 <= Arg[z] < π) || (-π < Arg[z] <= 0)], 
 Filling -> Bottom, FillingStyle -> Directive[Opacity[0.7], Gray], 
 Mesh -> {Range[0, 3, .5], 10}, 
 MeshFunctions -> {Abs[#2] &, Arg[Sin[#1]] &}, PlotRange -> {0, Pi}, 
 ColorFunction -> "CyclicLogAbsArg", PlotPoints -> 50, 
 ViewPoint -> {1.8, 1.75, 2.25}]

enter image description here

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  • $\begingroup$ In fact, the Abs[] is unnecessary here; all it's doing is turning the surface red. $\endgroup$ – J. M.'s ennui Feb 22 at 16:28

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