Useful conformal mappings

This post is to be a list of conformal mappings, so that I can get better at answering questions like “Find a conformal mapping from <this domain> to <this domain>”. The following Mathematica code is rough-and-ready, but it is designed to demonstrate where a given region goes under a given transformation.

whereRegionGoes[f_, pred_, xrange_, yrange_] := 

whereRegionGoes[f, pred, xrange, yrange] = 
 With[{xlist = Join[{x}, xrange], ylist = Join[{y}, yrange]},
  ListPlot[
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    Through[{Re, Im}[
     f /@ (#[[1]] + #[[2]] I & /@ 
      Select[Flatten[Table[{x, y}, xlist, ylist], 1], 
       With[{z = #[[1]] + I #[[2]]}, pred[z]] &])]]]]
  • Möbius maps - these are of the form \(z \mapsto \dfrac{az+b}{c z+d}\). They keep circles and lines as circles and lines, so they are extremely useful when mapping a disc to a half-plane. A map is defined entirely by how it acts on any three points: there is a unique Möbius map taking any three points to any three points (and hence any circle/line to circle/line). (Some of the following are Möbius maps.)
  • To take the unit disc to the upper half plane, \(z \mapsto \dfrac{z-i}{i z-1\)}
  • To take the upper half plane to the unit disc, \(z \mapsto \dfrac{z-i}{z+i}\) (the Cayley transform)
  • To rotate by 90 degrees about the origin, \(z \mapsto i \)z
  • To translate by \(a\), \(z \mapsto a+\)z
  • To scale by factor \(a \in \mathbb{R}\) from the origin, \(z \mapsto a \)z
  • \(z \mapsto exp(z)\) takes a vertical strip to an annulus - but note that it is not bijective, because its domain is simply connected while its range is not.
  • \(z \mapsto exp(z)\) takes a horizontal strip, width \(\pi\) centred on \(\mathbb{R}\) onto the right-half-plane.

Maps which might not be conformal

These maps are useful but we can only use them when the domain doesn’t include a point where \(f’(z) = 0\) (as that would stop the map from being conformal).

  • To “broaden” a wedge symmetric about the real axis pointing rightwards, \(z \mapsto z^\)2
  • To take a half-strip \(Re(z) > 0, 0 < Im(z) < \dfrac{\pi}{2}\) to the top-right quadrant: \(z \mapsto \sinh(z\))
  • to take a half-strip \(Im(z) > 0, -\frac{\pi}{2} < Re(z) < \frac{\pi}{2}\) to the upper half plane, \(z \mapsto \sin(z\))