1
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ComplexRegionPlot[Abs[z + 1 - I] <= 1, {z, -2, 1 + 2 I}, AxesOrigin -> {0, 0}, AspectRatio -> 2/3] ComplexRegionPlot[Arg[z - 1] <= 3 Pi/4, {z, -2, 1 + 2 I}, AxesOrigin -> {0, 0}, AspectRatio -> 2/3]

ComplexRegionPlot[ Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1, {z, -2, 1 + 2 I}, AxesOrigin -> {0, 0}, AspectRatio -> 2/3]

works fine, but maybe the boundaries need more points to refine it. enter image description here

But how would I get something this?

ComplexRegionPlot[ {Arg[z - 1] <= 3 Pi/4, Abs[z + 1 - I] <= 1, Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1}, {z, -2, 1 + 2 I}, AxesOrigin -> {0, 0}, AspectRatio -> 2/3]

enter image description here enter image description here

What I have tried

ComplexRegionPlot[ {Arg[z - 1] == 3 Pi/4, Abs[z + 1 - I] == 1, Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1}, {z, -2, 1 + 3 I}, GridLines -> {{-1, 0}, {0, 1, 2}}, PlotLegends -> "Expressions", AspectRatio -> 1]

ComplexRegionPlot[ {Arg[z - 1] <= 3 Pi/4, Abs[z + 1 - I] <= 1, Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1}, {z, -2, 1 + 3 I}, GridLines -> {{-1, 0}, {0, 1, 2}}, PlotLegends -> "Expressions", AspectRatio -> 1]

One shows too much, and one is missing the circle... enter image description here enter image description here

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Use PlotPoints and PlotRangePadding for the first case.

ComplexRegionPlot[{Arg[z - 1] <= 3 Pi/4, Abs[z + 1 - I] <= 1, 
  Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1}, {z, -2, 1 + 2 I}, 
 AxesOrigin -> {0, 0}, AspectRatio -> Automatic, 
 PlotRangePadding -> {{0.1, 0.1}, {0.1, 0.1}}, PlotPoints -> 50
 ]

enter image description here


Where the circle is not showing, change to an inequality:

ComplexRegionPlot[{Arg[z - 1] == 3 Pi/4, Abs[z + 1 - I] < 1, 
  Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1}, {z, -2, 1 + 3 I}, 
 GridLines -> {{-1, 0}, {0, 1, 2}}, PlotLegends -> "Expressions", 
 AspectRatio -> 1, PlotPoints -> 50]

enter image description here


For the last one:

ComplexRegionPlot[
 {
  Arg[z - 1] <= 3 Pi/4,
  Abs[z + 1 - I] <= 1,
  Arg[z - 1] <= 3 Pi/4 && Abs[z + 1 - I] <= 1
  },
 {z, -2, 1 + 3 I}
 , Frame -> True
 , GridLines -> {{-1, 0}, {0, 1, 2}}
 , PlotLegends -> "Expressions"
 , AspectRatio -> Automatic
 , PlotRange -> {{-2, 1}, {0, 2}}
 , PlotRangePadding -> {{0.1, 0.1}, {0.1, 0.1}}
 ]

enter image description here

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ComplexRegionPlot[{Arg[z - 1] <= 3 Pi/4 && 
    Abs[z + 1 - I] <= 1}, {z, -2, 1 + 3 I}, 
  GridLines -> {{-1, 0}, {0, 1, 2}}, PlotLegends -> "Expressions", 
  AspectRatio -> 1];
ComplexContourPlot[{Arg[z - 1] == 3 Pi/4, 
   Abs[z + 1 - I] == 1}, {z, -2, 1 + 3 I}];
Show[%%, %]

enter image description here

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