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I'm working with the text Complex Var. and Applications by Ward and Churchill which has a table of mappings in the back. The best I can do is to compose different mappings (not sure if this is how these problems are generally attacked).

From what I can see (and I am probably way off base) I can either map a closed rectangle along and above the x-axis to a half-ellipse, or map a half-annulus (centered at the origin) to a half-ellipse. I'm not sure how I could map the unit circle into either of those two, the box or annulus.

Thanks in advance, any help is appreciated.

PS if this is in the wrong section please move it.