= Solution
{c}
Use the <univalent functions> on $D$, namely the <univalent functions> $F:D\to\mathbb C$. Each has nonzero <derivative>, so its <image> $F(D)$ is a <domain> <conformally equivalent> to $D$, and is therefore <simply connected>. This <image> is proper: otherwise $F^{-1}:\mathbb C\to D$ would be a bounded nonconstant <entire function>, contrary to the <Liouville theorem>.
Conversely, every proper <simply connected domain> $G\subset\mathbb C$ is $F(D)$ for some such $F$, by taking the inverse of a <Riemann mapping theorem> bijection. Thus these <functions> parameterize all proper <simply connected domains> by their images. The <complex plane> itself is the exceptional <simply connected domain> and must be listed separately.
The parameterization has redundancy: two <univalent functions> $F_1,F_2$ have the same <image> if and only if $F_2=F_1\circ\phi$ for an <automorphism of the unit disk> $\phi$. All such <automorphisms> have the form
$$
\phi(z)=e^{i\theta}\frac{z-\alpha}{1-\overline\alpha z},\qquad |\alpha|<1,\quad\theta\in\mathbb R.
$$
If an interior point $w_0\in G$ is prescribed, the normalization $F(0)=w_0$ and $F'(0)>0$ real removes this redundancy. Equivalently one can describe these maps through their convergent <Taylor series> on $D$, imposing <univalence> on the resulting <holomorphic functions>.
Back to article page
