How to show the map of a complex function ( plane z incl the basic square grid to plane w?

Question

I want to make a presentation of the maps of the standard complex functions, as well as some mobius transformations and special complex functions.
I want to present it exactly like it is done in the answer from Michael E2 to this question: How can I improve my code for visualizing a complex map?
With one difference however. I want to show the effect of a complex function on the basic grid containing of squares 1-by-1, instead of the grid of 'polygons in circles' in that answer. I am lost in how to do that.
How should I do this? ( How can I apply a complex function to the basic grid? )

cvgmt · Accepted Answer

Block[{z = x + I y}, 
  ParametricPlot[ReIm[(z + 1)/(z - 1)], {x, -6, 6}, {y, -6, 6}, 
   MeshFunctions -> Automatic, Mesh -> {Range[-4, 4], Range[-4, 4]}, 
   Axes -> False, PlotStyle -> None, Axes -> False, MeshStyle -> Blue,
    BoundaryStyle -> Directive[Dotted, Red, Thick], PlotPoints -> 50]];
Block[{z = x + I y}, 
  ParametricPlot[
     ReIm[(z + 1)/(z - 1)], {x, #1, #1 + 1}, {y, #2, #2 + 1}, 
     MeshFunctions -> Automatic, 
     Mesh -> {Range[#1, #1 + 1], Range[#2, #2 + 1]}, 
     MeshShading -> {{None, None}, {None, Yellow}}, Axes -> False, 
     PlotStyle -> None, Axes -> False, MeshStyle -> Blue, 
     PlotPoints -> 50] & @@@ {{2, 2}, {3, 2}, {4, 2}, {3, 3}, {3, 1}}];
Show[%, %%, PlotRange -> {{-1, 3}, {-2, 2}}]

Block[{z = x + I y}, 
  RegionPlot[
   RegionUnion[
    ParametricRegion[
       ReIm[z], {{x, #1, #1 + 1}, {y, #2, #2 + 1}}] & @@@ {{2, 2}, {3,
        2}, {4, 2}, {3, 3}, {3, 1}}], BoundaryStyle -> LightRed, 
   PlotStyle -> LightGreen, PlotRange -> All]];
Block[{z = x + I y}, 
  RegionPlot[
   RegionUnion[
    ParametricRegion[
       ReIm[(z + 1)/(z - 1)], {{x, #1, #1 + 1}, {y, #2, #2 + 
          1}}] & @@@ {{2, 2}, {3, 2}, {4, 2}, {3, 3}, {3, 1}}], 
   BoundaryStyle -> Red, PlotStyle -> Yellow]];
GraphicsRow[{%%, %}]

a = Block[{z = x + I y},
   Show[ParametricPlot[ReIm[z], {x, -6, 6}, {y, -6, 6}, 
     MeshFunctions -> Automatic, Mesh -> {Range[-5, 5], Range[-5, 5]},
      Axes -> False, PlotStyle -> None, Axes -> False, 
     MeshStyle -> Blue, 
     BoundaryStyle -> Directive[Dotted, Red, Thick], PlotPoints -> 50],
    RegionPlot[
     RegionUnion[
      ParametricRegion[
         ReIm[z], {{x, #1, #1 + 1}, {y, #2, #2 + 1}}] & @@@ {{2, 
         2}, {3, 2}, {4, 2}, {3, 3}, {3, 1}}], BoundaryStyle -> Red, 
     PlotStyle -> Yellow], PlotRange -> {{-6, 6}, {-6, 6}}]];
b = Block[{z = x + I y},
   Show[ParametricPlot[ReIm[(z + 1)/(z - 1)], {x, -6, 6}, {y, -6, 6}, 
     MeshFunctions -> Automatic, Mesh -> {Range[-5, 5], Range[-5, 5]},
      Axes -> False, PlotStyle -> None, Axes -> False, 
     MeshStyle -> Blue, 
     BoundaryStyle -> Directive[Dotted, Red, Thick], PlotPoints -> 50],
    RegionPlot[
     RegionUnion[
      ParametricRegion[
         ReIm[(z + 1)/(
          z - 1)], {{x, #1, #1 + 1}, {y, #2, #2 + 1}}] & @@@ {{2, 
         2}, {3, 2}, {4, 2}, {3, 3}, {3, 1}}], BoundaryStyle -> Red, 
     PlotStyle -> Yellow], PlotRange -> {{-2, 3}, {-2, 2}}]];
GraphicsRow[{a, b}]

How to show the map of a complex function ( plane z incl the basic square grid to plane w?

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