Holomorphic logarithm of an open subset

Definition

Suppose $U\subset \mathbb{C}$ is an open subset bounded away from the origin (in other words, its closure does not contain the origin). A holomorphic logarithm of $U$ is an open subset $V\subset \mathbb{C}$ such that the exponential map $z\mapsto g(z)$ gives a biholomorphic mapping from $V$ to $U$.