  The Arens-Michael functor in noncommutative geometry is an analogue of the
analytification functor in algebraic geometry: out of the ring of "algebraic
functions" on a noncommutative space it constructs the ring of "holomorphic
functions" on it. In this paper, we explicitly compute the Arens-Michael
envelopes of the Jordanian plane and the quantum enveloping algebra
$U_q(\mathfrak{sl}(2))$ of $\mathfrak{sl}(2)$ for $|q|=1$.
