Planar logarithm factorization ($C(K,{\mathbb{C}}^{\ast })=\exp C(K)\times ⟨z-a_U:U\text{ bounded complementary component}⟩$)

Choose $a_U$ in each bounded component of $\mathbb{C}\setminus K$. Every nonvanishing continuous function on $K$ extends nonvanishingly to a neighborhood. Use finite-hole neighborhoods of a planar compact set within that neighborhood, and subtract its finite list of winding obstructions by multiplying by $\prod (z-a_U{)}^{-k_U}$. The continuous logarithm lifting criterion gives the remaining exponential. Uniqueness follows by extending a purported logarithm of such a finite product to a smaller neighborhood: a small multiplicative error has a local logarithm, whereas its winding around the $U$th hole is $k_U$. Thus all $k_U$ vanish. The quotient by exponentials is the free abelian group on the bounded complementary components; each element has finite support.