Can you verify this solution?
The question asks for a biholomorphic mapping of the domain $D=\{z \in \mathbb{C} : |z| < 1, \text{Re}(z) > 0, \text{Im}(z) > 0\}$ to the upper half-plane $P=\{z \in \mathbb{C} : \text{Im}(z) > 0\}$.
I first attempted to map the quarter-disk to the half-plane by mapping the circular arc of the quarter-disk to the line through the real axis by the transformation $$T_{1}(z) = \frac{z-i}{z} \cdot \frac{1}{1-i} = \dfrac{(1+i)z + (1-i)}{2z},$$ but this mapped it to the lower half-plane, so I multiplied by $e^{i \pi} = -1$ to rotate it to the upper half-plane, resulting in the map $$T(z) = \dfrac{(1+i)z + (1-i)}{-2z}.$$
That is, $T$ should map $D$ to $P$.
Is this correct? I found this question in a qualifying exam for a different school than mine, but there is no attached answer key.