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[
Transpose@
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,