Question: Find a conformal map from the exterior of the closed unit disk to the unit disk. Also, prove that it is indeed a conformal map (bijective and holomorphic along with its inverse).

I missed that the two days we covered conformal mapping due to chaos in my life. I have been trying to find worked out examples and have been reading up to get some insight, but I am lost since I can't find something that details a complete worked out problem.

p.s. If anyone knows a great resource that would help me out to further my grasp of conformal maps in complex analysis it'd be much appreciated if you could disclose this.


If you are working on $\mathbb{C}$, there is no such map (the exterior of the disc is not simply connected). If you're working on the Riemann sphere, investigate the mapping properties of $1/z$.

  • $\begingroup$ Like I said I was not there for any of the conformal maps discussion. I am good at teaching myself though. I did find questions on stack exchange where there were maps from the exterior of the disk to other domains such as exterior of an ellipse and I saw no discussion of Riemann sphere. The mapping I was given was f: C/D-->D (the "D" for C/D has a "-" above it which I know is the notation for what I had written.) $\endgroup$ – BrainDeadSenior Apr 15 '14 at 12:50
  • $\begingroup$ @BrainDeadSenior: Calling conformal maps "nonsense" is not a healthy attitude. Is the exterior of an ellipse simply connected? $\endgroup$ – mrf Apr 15 '14 at 12:52
  • $\begingroup$ I was editing my post I hit enter without realizing it...Relax, I don't find this or any aspect of mathematics to be nonsense it is just frustrating when you missed the entire discussion and your professor and the assigned text are both not very helpful. Which is why I have been hunting for resources and watching videos online. $\endgroup$ – BrainDeadSenior Apr 15 '14 at 12:55
  • $\begingroup$ $\bar{\mathbb{C}}$ is a common notation for the Riemann sphere $\endgroup$ – mrf Apr 15 '14 at 12:57
  • $\begingroup$ I know that notation. I'm saying the line is solely above the unit disk notation: "D." However, the sentence lays our "f" I was just trying to clarify. There is no abbreviation for Riemann sphere in the provided hand-out unless there was a typo that the professor made. Regardless, what is confusing me with your point is how there are other posts on this site that map from exterior of the disk to other domains and you are stating that there is no such map for the question I am asking due to exterior of disk not being simple connected. @mrf (unless I misunderstood you) $\endgroup$ – BrainDeadSenior Apr 15 '14 at 13:07

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