Solution
= Solution
The domains are not <conformally equivalent>. If $F:\mathbb C\to\mathbb H$ were a conformal equivalence, composing it with a <Cayley transform (hyperbolic geometry)> from the upper half-plane $\mathbb H$ to the unit disc would produce a bounded nonconstant <entire function>. This contradicts the <Liouville theorem>, proving that the <complex plane is not conformally equivalent to the upper half-plane>.