Find a conformal mapping from the complement of a half-line to the open unit disc

Question

I have a cut-plane, which is the complement of the half-line which starts at $z_0$ and makes and angle of $\theta$ with the positive real axis i.e. $\mathbb{C} \backslash\{z \in \mathbb{C}: z = z_0 + re^{i\theta}, r>0 \}$. I wish to find a conformal mapping $f$ which maps this set to the open unit disk $\{z \in \mathbb{C}: |z| < 1\}$.

Is there some general way of doing this?

Can it be done for specific "easy" cases such as when the half line the set complements lies on the imaginary or real axes only, for example the half line starting at $-1+0i$ which makes an angle of $\pi$ with the positive real axis?

Answer 1

The conformal map can be decomposed into three steps.

1. Rotate and translate the cut plane so that the cut is the negative real axis: $z\mapsto-(z-z_0)e^{-i\theta}$.
2. Apply the map $z\mapsto\sqrt z$. This turns the cut plane into the right half-plane.
3. Apply $z\mapsto\frac{z-1}{z+1}$ to map the right half-plane to the unit disc.