I would like to know how one can find a conformal map that maps a given set onto another (not $\mathbb D\rightarrow\mathbb D$ or $\mathbb H\rightarrow \mathbb H$ since those are clear).
For example, if the question is to
- find a conformal map that maps the upper half of $\mathbb D$ to the second quadrant of the plane.
- find a conformal map that maps the upper half of $\mathbb D$ to $\mathbb H$
How should one proceed? I've heard one can fix 3 points and this determines the mapping but how to find to where they should be mapped
NB: $\mathbb D=D(0,1)$ the unit disk and $\mathbb H=\{z\in\mathbb C:Im(z)>0\}$
Best Answer
Strategies
There is not a single general method for finding these mappings. Instead, we can learn several "building blocks" and then assemble these to construct most mappings we might want. The following tricks are enough to cover your example questions:
A couple of bonus tricks just for fun:
Solving your specific proposed examples
Upper half disc to 2nd quadrant of the plane: Using trick (4), we see it's sufficient to find mappings from each region to $\mathbb{D}$. Trick (3) shows how to map from upper half disc to $\mathbb{D}$, since the upper half disc is bounded between the unit circle and the line $\text{Im}(z) = 0$. Trick (2) shows how to map from the 2nd quadrant to $\mathbb{D}$.
Upper half disc to upper half plane: Use trick (3) in the same way as above.