This problem is from Ahlfors; Chapter 3, Section 5, Page 97, Problem 5 .

Statement: Map the inside of the right-hand branch of the hyperbola on the disk so that the focus corresponds to and the vertex to .

Recall that we can describe this hyperbola with center , focus , and a real constant as from the basic definition of a hyperbola. For , we can derive where . Let then . Set where is specified in the hypothesis. Then which has center , vertex , and focus .

Call the open region to be mapped into the disk . My first (failed) attempt was to try and find a point symmetric to the focus and outside so that I could map , , and . However, no such symmetric point exists that I could identify.

Ahlfors also discusses, on p. 90, the use of level curves on the components of the map , but I don't see how I can use that either.

Ideas and / or a solution are both welcome at this point. I appreciate your help.