It is a well known theorem that any doubly connected domain can be conformally mapped onto an annulus.
Consider the simpler version :
Suppose $D$ is a bounded domain whose boundary is two non-intersecting circles. Then $D$ can be conformally mapped onto an annulus.
I believe that the proof of this should be easier, that the conformal map would be just a linear fractional transformation. However, I'm having trouble constructing it. Could someone help?