I'm attempting a complex analysis exercise from early in my class' course, just after we've been taught mobius transformations. The rest of the problem sheet was relatively simple, but I'm absolutely stumped as to where to begin here. Can anyone offer guidance?
Define
$$U^1 := \{(z_1, z_2) \in \mathbb{C} \times \mathbb{C} : \text{Im}(z_2) > |z_1|^2\}$$
a) Show there are maps $f_1, f_2$ such that $(z_1, z_2) \to ( f_1(z_1), f_2(z_1, z_2))$ is a bijection between the set $U^1$ and $$\{(w_1, w_2) \in \mathbb{C} \times \mathbb{C} : |w_1|^2 + |w_2|^2 < 1 \}. $$