# complex region plot - tweaking

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.

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]

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...

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
]


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]


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}}
]


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[%%, %]