Complex logarithm

Remarks

Let $|z|$ be the modulus of $z$, let $\ln$ be the real natural logarithm, and let $\theta$ be an argument of $z$. Then the logarithms of $z$ are exactly the numbers $\ln |z| + i (\theta + 2 \pi k)$ with $k \in \mathbb{Z}$. The number $0$ has no logarithm, because $e^w \neq 0$ for each $w$. So $\log$ is a multivalued function on $\mathbb{C} \setminus \{0\}$. Its principal branch is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, with $-\pi < \operatorname{Arg} z \le \pi$. For example, the logarithms of $-1$ are $i \pi (2k + 1)$ with $k \in \mathbb{Z}$, and $\operatorname{Log}(-1) = i \pi$.