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Patrick Stevens

Former mathematics student at the University of Cambridge; now a software engineer.

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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[

   [email protected]

    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 . 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,
  • To take the upper half plane to the unit disc, (the Cayley transform)
  • To rotate by 90 degrees about the origin,
  • To translate by ,
  • To scale by factor from the origin,
  • 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.
  • takes a horizontal strip, width centred on 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 (as that would stop the map from being conformal).

  • To “broaden” a wedge symmetric about the real axis pointing rightwards,
  • To take a half-strip to the top-right quadrant:
  • to take a half-strip to the upper half plane,