### 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[
Transpose@
Through[{Re, Im}[
f /@ (#[] + #[] I & /@
Select[Flatten[Table[{x, y}, xlist, ylist], 1],
With[{z = #[] + I #[]}, 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, $$z \mapsto \sin(z$$)