Plotting a complex valued function over a circular region

I would like to apply some complex valued function to some region in the plane, say, a circle of radius $R$ centred at $k$.

How can I do this?

Answer

Under the interpretation of OP's question as

How do I apply the transformation $w=f(z)$ to a region (e.g. a disk) in the complex plane?

I'd say ParametricPlot[] (which now incorporates the functionality from the old Graphics`ComplexMap` package) would be what you can use:

With[{f = # + 1/# &, center = 1/3 + 3 I/2, radius = 4/3}, 
 ParametricPlot[
       Through[{Re, Im}[f[center + r Exp[I θ]]]], {r, 0, radius}, {θ, -π, π},
               PlotPoints -> 45, PlotRange -> All]]

transformation of a disk

Comments

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