#### QUESTION

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]]
```

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