Find holomorphic mapping from domain onto unit disc $|z|<1$ I am trying to find a holomorphic mapping from some domain onto $\{z: |z|<1\}$. 
The problem in the textbook was finding the holomorphic mapping from the domain (1) $\{z: |z+1| > 1, |z+2| <2\}$, (2) $\{ z: |z|<1, |z-1|>1\}$  onto $\{z : |z|<1\}$. 
First I know the holomorphic map is a function with $z$ variables and try to cook up with polynomials, but this seems not good since the regions are restricted by $|z|<1$. 
Is there any systematic procedure in general? 
 A: What at about the Cayley transformation?  It's $f(z)=\dfrac{z-i}{z+i}$.  The upper half plane is mapped to the unit disc.
In general, keep track of where generalized circles go.  You can do that by checking three points.  Then check where the enclosed region goes by using a test point.
For instance, the Cayley transform takes $\infty,  1, -1$ to $1, -i, i$ respectively.  And $i$ to $0$.
Now try to compose maps (Mobius transformations) that do the appropriate things.  But you will need another tool to get this to work.  See the other answer.
A: You will need Möbius transformations, and something else. Möbius transformations (also known as fractional linear transformations) are flexible tools that will map circles to circles or straight lines, but they will at all times preserve angles, which prevents them from being your only tool.
Your first domain lies between two circles that touch each other at $0$. A Möbius transformation that sends $0$ to $\infty$ will send the two circles to parallel lines. Arrange so that your domain is mapped to the parallel strip between the $x$-axis and the line $y=\pi$. Now use the complex exponential function, and check that this parallel strip is mapped to the upper half plane. Finally, map the upper half plane to the unit disc with another suitably chosen Möbius transformation.
For the second domain, you have two circles that intersect in two different points. Choose a Möbius transformation that sends one point to $0$ and the other to $\infty$, and such that your domain is sent to a sector in the upper half plane with one leg along the positive real axis. For a suitably chosen $\alpha$, the function $z\mapsto z^\alpha$ will map this sector to the upper half plane. Finally, map the upper half plane to the unit disc, as in the first case.
