How does one define the periods that appear in this question and show they are conformally equivalent? Are the details worked out in a textbook somewhere?
Presumably we do something like take the harmonic function that's zero on the inner circle, 1 on the outer circle, find the multivalued harmonic conjugate, and integrate the resulting multi-valued analytic function around a path that loops around the interior circle once. But it's been a while since I've done complex analysis and I get a little nervous when multi-valued functions start popping up. A formal reference (or write-up) would be very much appreciated.
It seems details can be found in Nehari's Conformal Mapping, Chapter 1, section 9.
Perhaps a better reference is Chapter 6, section 5 of Ahlfors' Complex Analysis.