While preparing for a test I am encountering a lot of problems of type:
Find a conformal map from half-plane/unit disk to half-plane/unit disk such that point $z_0$ goes to $w_0$ and derivative of map at point $z_0$ has an argument equal to $\alpha$.
Firstly, what is a reason to give such problems? Are such calculations extensively used in some physics/applied math problems?
Secondly, how to solve such problems? I can try to move a this problem to problem of finding a specific upper halfplane/disk automorhisms and start solving system of equations: $\phi(z_0)=w_0\; \phi'(z_0) = ke^{i\alpha}$ for $\phi$ having a usual disk/halfplane automorphism form.
I had found an answer for disk variant in textbook
But I don't know how to get this formula, when I tried, calculations got very messy and I could not move forward without constantly making mistakes and involution gets into my way too much. I can sometimes solve a system for a concrete values, but usually I bury myself in calculations. What would be a less computation heavy approach?