\begin{definition}[Definition:Natural Logarithm/Complex/Principal Branch]
The principal branch of the complex natural logarithm is usually defined in one of two ways:
:$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointr 0 {2 \pi}$
:$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointl {-\pi} \pi$
It is important to specify which is in force during a particular exposition.
\end{definition}