Definition 31.5.4 (Principal Logarithm).label Let $U = \complex \setminus \bracs{z \in \real|z \le 0}$, then there exists a unique mapping $\ell: U \to \complex$ such that:

1. (1)

   $\ell$ is a branch of the complex logarithm.

2. (2)

   For each $re^{i\theta}\in U$, $\ell(r^{i\theta}) = \ln r + i\theta$.

The function $\ell$ is the principal logarithm on $U$.