Conformal mapping of the disk $| z | <R_1$ to disk $| w | <R_2$

Question

could you help me with the following please:

Find the conformal mapping of the disk $| z | <R_1$ to disk $| w | <R_2$ such that $w (a) = b, Arg(w´(a)) = \alpha $, $(| a | <R_1, | b | <R_2)$.

I have tried the following, but it does not match the solution that comes in the book: We go to the unit circle with $w_1=\frac{z}{R_1}$, and then to the radio $R_2$ with $w_2=R_2 w_1$ and the transformation we are looking for is then its composition, I do not know how to make the conditions are fulfilled. I would greatly appreciate your help

Answer 1

Hint : What are the complex automorphisms of the disk ? Could they be used to fulfill the conditions ?