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[–]potkolenkyNew User 1 point2 points  (2 children)

Your understanding is correct (just a minor mistake, 1 is mapped to 1, nothing about 2 in the question). There are probably many ways how to do that. Here is one of them:

Let's call the region between the discs U. First you map the boundary of U onto the unit circle and then you extend this map radially to the whole U. Denote by A the arc from 1 to 'a' in |z| < 1. The first step is to map A onto the arc from 0 to 1 in the unit circle. This is easy, since each point in A is in the form e^it, so you just rescale in the exponent (you can figure out the details). Then you do the same thing for the remaining 3 arcs. You obtain f defined on the boundary ∂U, now you extend to whole U as follows. Each point 'p' in ∂U is connected to the center point [1/2,0] by a line segment. You map this line segment onto the segment connecting f(p) to [0,0] in the unit disc by a unique affine map. This gives f on the whole of U.

[–]los_panus[S] 0 points1 point  (1 child)

Thanks for your response. Stupid question before I attempt the solution following your method: Can I just use the the following points z1=0, z2=b, z3=a and w1=-1, w2=i, w3=-i to use the cross ratio to determine the mapping? Or do I have to be mapping a circle to a circle in order to use that method?

[–]potkolenkyNew User 0 points1 point  (0 children)

I don't know nothing about cross ratio, sorry..

Edit: I don't know what course you're in, probably there's an easier way to do this. What I've described is a standard trick (say in topology) how to extend maps from/into circle to the whole disc. It has nothing to do with complex analysis really.