Thread: Conformal Mapping to Unit Disc

Problem: Find a conformal map from the set A = { z = x + iy in C : x^2 + y^2 > 1, x > 0} to the unit disc {z in C : |z| < 1}.

In class we've been through some examples of this form where we construct a conformal map as a composition of mobius maps and other analytic functions. (exp(z), z^n, etc). In each example we used such compositions to map the given sets into the upper half plane { Im(z) > 0} or the half plane {Re(z) > 0}, and then it is easy to map these conformally into the unit disc. So for this set A I imagine I need to do the same thing, but I can't find the right map(s). Help please!

The inverse of the stereographic projection sends A to half of the upper half Riemann sphere, reflecting via the xy plane sends to the lower half( like f(z)=1/z), stretch the 1/4 sphere to the 1/2 sphere( like f(z)=z^2), then sends back via the stereographic projection.

Thanks. I think I may have found a simpler composition though, can you see if there is anything wrong with it?

On the set A define f(z) = (z-1)/(z+1). This maps A into half of the unit disc (the half with Re(z) > 0). Now simply define g(z) = z^2 on this half-disc and we have a conformal map from A to the whole disc, namely h(z) = g(f(z)).