Try to visual this set. Use several values to to motive you. If then is the cut-plane at the non-positive axis (cut along ). The square root map (where the branch of the square root is choose along the same cut i.e. principle branch) will map this set to i.e. the right half-plane. But you want the upper-half plane to rotate your picture one-quater turn i.e. will do the trick. This gives the conformal mapping . Thus, in general if we want to shrink (or expand) to we should use as you correctly said. But this only gives us the right half plane, therefore, we need . Good job .