Find an analytic function
$f: Ω\to\mathbb{C}$
which takes the semicircular region
$Ω =\{z\in\mathbf{C}:|z|<1,\text{Im} z >0\}$ one-to-one and onto the unit ball
$B_1(0) =\{z\in\mathbf{C}:|z|<1\}.$
My idea is let $f(z)=z^2$, but it seems the real segment [0,1) is missing. Is there any other function satisfying the requirement?
The mapping $$h: \Omega \longrightarrow Q_1, z \mapsto i \ \frac{1-z}{1+z}$$ maps the domain $\Omega$ conformally onto the first quadrant $Q_1 := \{ z = x + iy \in \mathbb{C} \ | \ x, y > 0 \}$. Subsequently, the mapping $z \mapsto z^2$ sends $Q_1$ conformally onto the upper half plane $UH := \{ z = x + iy \ | \ y > 0 \}$. Finally, the mapping $$g: UH \longrightarrow B_1(0), z \mapsto \frac{z-i}{z+i}$$ maps $UH$ conformally onto the unit disk $B_1(0)$. Hence the composed mapping $$f := g \circ z^2 \circ h: \Omega \longrightarrow B_1(0)$$ is the desired mapping you are looking for.