Conformal mappings

[sans_galois]

I have two questions:

1. Find a bijective conformal mapping from G={z in C : |z|<2 and |z-1|>1} onto the open unit disc D.

I'm not really sure where to begin. This set G is more complicated than any of the examples we have of this sort of process; all of our examples have been simply connected.

2. Show that f(z)=(z2 - 1)/(z2 + 1) is a bijective

Comments

[st_rev]

1. Find a Mobius transformation that sends the "bad" point to infinity, figure out what the image of that looks like, then see if you can send that to the unit disc.

2. (w-1)/(w+1) is another Mobius transformation; they're very well-behaved geometrically.

Generally: read up on Mobius transformations, and you ought to find both problems pretty straightforward.

[sans_galois]

I figured out part one. I was fooling myself by drawing a picture with solid lines rather than dashed ones, so I kept the set wasn't simply connected... Whoops

And as I started typing up my confusion for part two, I actually figured out the first part, haha. I've tried to work through the second part under the assumption I got the first (which I now have), but it's still a bit unclear. It seems as though I want the the map g=if-1, but writing it out and then showing that tan(g(z))=z....

[sans_galois]

Sometimes pictures help. And sometimes pictures lead me to believe that sets are not simply connected when they are

:|