I know this is a bit vague and there's no end to the conformal maps, but I'm just curious which ones you've memorized out of sheer usefulness, particularly if you work in an area related to complex analysis. Which ones should one be able to derive as well?
Which conformal maps should one have memorized?
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1Something I never really internalized was the fact that Möbius transformations correspond to [choices of three points](http://en.wikipedia.org/wiki/Möbius_transformation#Specifying_a_transformation_by_three_points). It seems like that could be really useful. – 2012-06-16
1 Answers
There aren't that many to memorize, and you should be able to derive them all. Know how to use linear fractional transformations, including regular linear transformations like rotations and translations. Know what the exponential and logarithm maps do. Know what taking powers and roots looks like. You should be especially familiar with how to map the unit disk to the upper half plane and back.
You should also know the Schwarz-Cristoffel and Riemann Mapping theorems.
The best way to get comfortable with conformal mapping is simply to do a lot of problems. I know Churchill's book has many problems on this topic, and so should any other standard text. There are a very small number of speciality tricks (e.g. mapping an ellipse to a disk) and you just pick those up from seeing them used. The vast majority of conformal mapping you do (I'm assuming this is for coursework) is going to be finding clever ways to compose the basic maps I listed above. This is a skill you pick up through practice.
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0I'm pretty new to conformal mappings, @Potato, but I wanted to ask you: a lot of these mappings that should be memorized aren't necessarily conformal, right? For instance, I think the power map, Z^2, doubles the angles of the inputs / pre-image points, so angles between curves are not preserved. And the exponential map is 2(pi)(i) periodic and so is not injective - and so is not conformal either, since conformal mappings are defined to be one-to-one, onto mappings, with the additional requirement that the mappings preserve angles *and* the angles orientation. What are you thoughts? – 2014-12-07
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0It's a little confusing, as a beginner like myself, when I read books and online material, and I started to make the mistake that...every transformation that I'm learning about...is a conformal mapping, when that's probably not the case. Thanks in advance, @Potato.. – 2014-12-07
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0For instance, is a reflection mapping a conformal mapping?A reflection operator reverses orientation. What about inversion with respect to a circle? I'd like to think that we are learning a lot of "linear transformations", but they are not necessarily Mobius transformations, which are also known as Linear Fractional Transformations. But, they seem to all get put into the same chapter in a book, the same topic on a course webpage, etc... – 2014-12-07
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0Or, perhaps those mappings that I've mentioned that aren't conformal...can always be *made* conformal, when restricted to a certain domain in the complex plane, say, a certain disk away from certain points, a certain strip (vertical or horizontal), etc... – 2014-12-07
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0Another guess would be...perhaps we don't actually care all that much about whether the mapping is conformal, but rather we just care that we "get the job done". Say, we have a difficult domain to start with, e.g., the upper half plane minus the upper half of the unit disk, and we should map that region to the disk first, before proceeding with solving that problem (perhaps it's a boundary value problem in PDE). but, this mapping from that region to the disk need not be conformal, if the problem doesn't specify, right? – 2014-12-07
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0(or, in these types of problems, perhaps it's *always* implied that any intermediate and final mappings that we apply should absolutely be conformal...) – 2014-12-07
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0And, of course, rotations, dilations, and translations preserve angles and the angles' orientation. But, I'm not too sure about the inversion mapping, 1/z, even though that's one of the mappings that give us the general form of a Mobius transformation, az+b / cz+d, since inversion is somewhat like a reflection operator, which would reverse orientation. – 2014-12-08
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0@LebronJames That's a lot of questions! The maps I indicated are conformal when restricted to appropriate domains. A reflection is not even holomorphic. If by inversion with respect to a circle, you mean something like $1/z$, that's conformal when properly restricted. Generally, if you're trying to solve PDE problems (I have in mind Laplace's equation), you need the mapping to be conformal (look at the details in whatever examples you have and you'll see why). – 2014-12-08
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0@LebronJames It sounds like you would benefit from reading a good textbook. I recommend Stein and Shakarchi's *Complex Analysis*. – 2014-12-08
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0ok, thanks so much, @Potato. After all my questions that I posted for you, I watched a few online lectures on conformal mappings and, specifically, Mobius transformations. That helped a lot, but I'll definitely try to consult Stein's book too. Have a great night :) – 2014-12-08
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0and yea, thanks a ton for reading all of my questions haha :) our prof. this semester just expected us to learn it on our own and went right for the proof of the Riemann Mapping Theorem. i have a lot of background to fill in.. – 2014-12-08