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45 lines
2.7 KiB
Markdown
45 lines
2.7 KiB
Markdown
---
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lastmod: "2021-09-12T22:47:44.0000000+01:00"
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author: patrick
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categories:
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- uncategorized
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comments: true
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date: "2014-04-07T00:00:00Z"
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math: true
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aliases:
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- /uncategorized/useful-conformal-mappings/
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- /useful-conformal-mappings/
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title: Useful conformal mappings
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---
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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.
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whereRegionGoes[f_, pred_, xrange_, yrange_] :=
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whereRegionGoes[f, pred, xrange, yrange] =
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With[{xlist = Join[{x}, xrange], ylist = Join[{y}, yrange]},
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ListPlot[
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Transpose@
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Through[{Re, Im}[
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f /@ (#[[1]] + #[[2]] I & /@
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Select[Flatten[Table[{x, y}, xlist, ylist], 1],
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With[{z = #[[1]] + I #[[2]]}, pred[z]] &])]]]]
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* 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.)
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* To take the unit disc to the upper half plane, \\(z \mapsto \dfrac{z-i}{i z-1\\)}
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* To take the upper half plane to the unit disc, \\(z \mapsto \dfrac{z-i}{z+i}\\) (the [Cayley transform][1])
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* To rotate by 90 degrees about the origin, \\(z \mapsto i \\)z
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* To translate by \\(a\\), \\(z \mapsto a+\\)z
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* To scale by factor \\(a \in \mathbb{R}\\) from the origin, \\(z \mapsto a \\)z
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* \\(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.
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* \\(z \mapsto exp(z)\\) takes a horizontal strip, width \\(\pi\\) centred on \\(\mathbb{R}\\) onto the right-half-plane.
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## Maps which might not be conformal
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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).
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* To "broaden" a wedge symmetric about the real axis pointing rightwards, \\(z \mapsto z^\\)2
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* To take a half-strip \\(Re(z) > 0, 0 < Im(z) < \dfrac{\pi}{2}\\) to the top-right quadrant: \\(z \mapsto \sinh(z\\))
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* 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\\))
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[1]: https://en.wikipedia.org/wiki/Cayley_transform#Conformal_map "Cayley transform Wikipedia page"
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