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

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? )

• See this demo demonstrations.wolfram.com/ConformalMaps Aug 28 '20 at 13:56
• This is another relevant demo demonstrations.wolfram.com/ConformalMapOfExponential Aug 28 '20 at 13:57
• Thank you very much for these links, I always check Wolfram Demos but didn't find them. - I hope that from studying the source code of these demos I can move on. The point is that it is not just the Conformal Mappings I want to demonstrate but basically all complex functions. Aug 28 '20 at 14:10
• Block[{z = x + I y}, ParametricPlot[ReIm[(z + 1)/(z - 1)], {x, -Pi, Pi}, {y, -2, 2}, Mesh -> 9, Axes -> False]] Aug 28 '20 at 14:12
• Thanks. Is it that simple? - You can put that in an answer. Aug 28 '20 at 14:20

 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}]


• Thank you. - Still a lot of work to do before I can make slides from this... Have you seen the cool layouts from the accepted answer at linked question mathematica.stackexchange.com/questions/156347/… ; all I want is a standard grid as source. Aug 29 '20 at 17:25