One way to construct conformal maps like this is to piece them together from simple building blocks. To do that, you need to have a "library" of known simple conformal maps. For example, you know that fractional linear maps take lines and circles to lines and circles. So a good place to start would be the map . This maps 2 to infinity, and takes the circles |z–1|=1 and |z|=2 to the lines re(z)=1/4 and re(z)=1/2. So it maps the region R between the circles to the strip .
Next, does our library of conformal maps include one that takes a strip to a half-plane? Yes it does! The exponential map takes the strip to the upper half-plane H.
So now all you need is an affine map to dilate and rotate to , and then the composition of the three maps will take R to H. The result is the map . (Of course, that's not the unique answer, because there are many conformal maps from H to itself.)