I'm currently working on a problem that asks to map the unit quarter disk from quadrant I to the upper half-plane, and then to use this mapping to find a harmonic function on the quarter disk, taking values 1 on the part of the boundary with |z| = 1 and taking the value 0 on the rest of the boundary of the quarter disk.
I think I computed the correct conformal mapping.
First, I applied the mapping $z^4$. This maps the quarter disk to the unit disk.
Then, I used the standard mapping from the UHP to the unit disk, and computed its inverse. Now I have a mapping taking the quarter disk to the upper half plane, as required:
$$z \to \frac{i(1+z^4)}{1-z^4}$$
I'm having trouble finding the harmonic function, though. I think that the part of the boundary of the quarter disk that is supposed to take the value 1 is mapped to the negative real axis. Ok, then I apply the harmonic $Arg$ function to the above mapping, and I should get $\pi$ as the output, along the negative real axis. Now I scale by 1/$\pi$ to get my boundary value of 1. The function is
$$z \to \frac{1}{\pi}Arg [\frac{i(1+z^4)}{1-z^4}]$$
However, I don't know how to make the rest of the boundary of the quarter disk take the value 0.
Thanks,
$z^4$ maps the (open) quarter disk on the disk minus the interval $[0,1)$. You should start with $z^2$, which takes the quarter disk to the half disk, and then $(z+1)/(z-1)$.