# How do I draw a conformal mapping from the z-plane to the w-plane

I am total beginner in Mathematica. Don't know a thing. I need to draw some diagrams for showing the conformal nature of map $$w=e^z$$.

I need to draw some contours in $$z$$-plane; e.g., horizontal line, vertical line, 45 degree line and unit circle. Then I need to obtain the mapped curves in $$w$$-plane where $$u$$ and $$v$$ are functions of $$x$$ and $$y$$.

Use MeshShading to fill the region.

expr1 = {x, y};
expr2 = With[{z = x + I*y}, E^z // ReIm // ComplexExpand];
ParametricPlot[#, {x, -4, 4}, {y, -4, 4},
MeshFunctions -> {#3 &, #4 &, #4 - #3 &, Sqrt[#3^2 + #4^2] &},
Mesh -> {{1}, {1}, {0}, {1}},
MeshShading -> {{{{Red, None}, {None, None}}, {{Red, None}, {None,
None}}}, {{{None, None}, {None, None}}, {{None,
None}, {None, None}}}},
MeshStyle -> {{Thick, Red, Opacity[1]}, {Thick, Green,
Opacity[1]}, {Thick, Blue, Opacity[1]}, {Thick, Yellow,
Opacity[1]}}, Axes -> False, PlotRange -> 4,
BoundaryStyle -> None,
LabelStyle -> {FontFamily -> "Times", Blue}, PlotPoints -> 80,
PlotStyle -> None] & /@ {expr1, expr2} // GraphicsRow


Simplify

expr1 = {x, y};
expr2 = With[{z = x + I*y}, E^z // ReIm // ComplexExpand];
ParametricPlot[#, {x, -4, 4}, {y, -4, 4},
MeshFunctions -> {#3 &, #4 &, #4 - #3 &, Sqrt[#3^2 + #4^2] &},
Mesh -> {{1}, {1}, {0}, {1}},
MeshStyle -> {{Thick, Red, Opacity[1]}, {Thick, Green,
Opacity[1]}, {Thick, Blue, Opacity[1]}, {Thick, Yellow,
Opacity[1]}}, Axes -> False, PlotRange -> 4,
BoundaryStyle -> None,
LabelStyle -> {FontFamily -> "Times", Blue}, PlotPoints -> 80,
PlotStyle -> None] & /@ {expr1, expr2} // GraphicsRow


Original

expr = With[{z = x + I*y}, E^z // ReIm // ComplexExpand]
xy = ParametricPlot[{x, y}, {x, -2, 2}, {y, -2, 2},
MeshFunctions -> {#1 &, #2 &, #2 - #1 &, Sqrt[#1^2 + #2^2] &},
Mesh -> {{1}, {1}, {0}, {1}},
MeshStyle -> {{Thick, Red, Opacity[1]}, {Thick, Green,
Opacity[1]}, {Thick, Blue, Opacity[1]}, {Thick, Yellow,
Opacity[1]}}, PlotPoints -> 50, FrameLabel -> {x, y},
PlotStyle -> None];
uv = ParametricPlot[expr, {x, -4, 4}, {y, -4, 4},
MeshFunctions -> {#3 &, #4 &, #4 - #3 &, Sqrt[#3^2 + #4^2] &},
Mesh -> {{1}, {1}, {0}, {1}},
MeshStyle -> {{Thick, Red, Opacity[1]}, {Thick, Green,
Opacity[1]}, {Thick, Blue, Opacity[1]}, {Thick, Yellow,
Opacity[1]}}, Axes -> False, PlotRange -> 8,
BoundaryStyle -> None, FrameLabel -> {u, v},
LabelStyle -> {FontFamily -> "Times", Blue}, PlotPoints -> 80,
PlotStyle -> None];
GraphicsRow[{xy, uv}]

• if you want a filled circle e.g, what is the code for that? – runner Jan 16 at 16:59
• @runner maybe more complexied. See the updated. – cvgmt Jan 17 at 1:50