Function Repository Resource:

ComplexMapVisualization

Visualize the behavior of conformal mappings in the complex plane

Contributed by: Duncan Pettengill

ResourceFunction["ComplexMapVisualization"][f]

returns the ImageTransformation of a given function f on the complex plane.

Details and Options

f can be any function that accepts a complex number as input and returns a complex number as output.

ResourceFunction["ComplexMapVisualization"] inherits options from ImageTransformation and Rasterize, which are passed to those functions during evaluation.

ResourceFunction["ComplexMapVisualization"] also has an "Image" option, so that mappings can be visualized on different base images. The default image is a grid of red and green lines.

With default option settings, the image is mapped using a DataRange and PlotRange of {{0,1},{0,1}}.

Examples

Basic Examples (3)

Visualize the sine function in the complex plane:

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Mappings on different sections of the complex plane can be visualized by specifying a different DataRange or PlotRange:

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Forward transformations can be created by using the inverse function:

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Scope (3)

Standard special functions can be visualized:

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General functions that have complex values as their domain and range can be visualized:

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$ResourceFunction["ComplexMapVisualization"][ Function[z, (z - 1)/(z^2 + 2)], PlotRange -> {{-3, 3}, {-3, 3}}, RasterSize -> 150]$

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Compiled functions can be visualized:

In[6]:=

$cf = Compile[{{x, _Complex}}, Sin[x] + x^2 - 1/(1 + x)]$

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Options (2)

The "Image" option can be set to visualize mappings on arbitrary images or graphics objects:

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Use the "Image" option to see the transformation of a polar grid:

In[9]:=

$ResourceFunction["ComplexMapVisualization"][ProductLog, PlotRange -> {{-1, 1}, {-1, 1}}, "Image" -> \!\(\* GraphicsBox[ {RGBColor[0, 1, 0], Thickness[0.01], CircleBox[{0, 0}, 0.], CircleBox[{0, 0}, 0.1], CircleBox[{0, 0}, 0.2], CircleBox[{0, 0}, 0.30000000000000004], CircleBox[{0, 0}, 0.4], CircleBox[{0, 0}, 0.5], CircleBox[{0, 0}, 0.6000000000000001], CircleBox[{0, 0}, 0.7000000000000001], CircleBox[{0, 0}, 0.8], CircleBox[{0, 0}, 0.9], CircleBox[{0, 0}, 1.], CircleBox[{0, 0}, 1.1], CircleBox[{0, 0}, 1.2000000000000002], CircleBox[{0, 0}, 1.3], CircleBox[{0, 0}, 1.4000000000000001], {RGBColor[1, 0, 0], TagBox[ConicHullRegionBox[{0, 0}, {{1, 0}}], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{(Rational[ 5, 8] + Rational[1, 8] 5^Rational[1, 2])^Rational[1, 2], Rational[1, 4] (-1 + 5^Rational[1, 2])}}, {{ 0.9510565162951535, 0.30901699437494745`}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{Rational[1, 4] (1 + 5^Rational[1, 2]), ( Rational[ 5, 8] + Rational[-1, 8] 5^Rational[1, 2])^Rational[ 1, 2]}}, {{0.8090169943749475, 0.5877852522924731}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{(Rational[ 5, 8] + Rational[-1, 8] 5^Rational[1, 2])^Rational[ 1, 2], Rational[1, 4] (1 + 5^Rational[1, 2])}}, {{ 0.5877852522924731, 0.8090169943749475}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{Rational[1, 4] (-1 + 5^Rational[1, 2]), ( Rational[ 5, 8] + Rational[1, 8] 5^Rational[1, 2])^Rational[ 1, 2]}}, {{0.30901699437494745`, 0.9510565162951535}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, {{0, 1}}], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{Rational[1, 4] (1 - 5^Rational[1, 2]), ( Rational[ 5, 8] + Rational[1, 8] 5^Rational[1, 2])^Rational[ 1, 2]}}, {{-0.30901699437494745`, 0.9510565162951535}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{-(Rational[ 5, 8] + Rational[-1, 8] 5^Rational[1, 2])^Rational[ 1, 2], Rational[1, 4] ( 1 + 5^Rational[1, 2])}}, {{-0.5877852522924731, 0.8090169943749475}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{Rational[1, 4] (-1 - 5^Rational[1, 2]), ( Rational[ 5, 8] + Rational[-1, 8] 5^Rational[1, 2])^Rational[ 1, 2]}}, {{-0.8090169943749475, 0.5877852522924731}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, NCache[{{-(Rational[ 5, 8] + Rational[1, 8] 5^Rational[1, 2])^Rational[ 1, 2], Rational[ 1, 4] (-1 + 5^Rational[1, 2])}}, {{-0.9510565162951535, 0.30901699437494745`}}]], "InfiniteLine"], TagBox[ConicHullRegionBox[{0, 0}, {{-1, 0}}], "InfiniteLine"]}}, PlotRange->1]\)]$