\begin{definition}[Definition:Natural Logarithm/Complex/Definition 1]
Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$.
The '''complex natural logarithm''' of $z \in \C_{\ne 0}$ is the multifunction defined as:
:$\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$
where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.
\end{definition}