Complex logarithm

Summary

A complex logarithm of a nonzero complex number $z$ is a complex number $w$ with $e^w = z$, see the definition. Each $z \neq 0$ has infinitely many logarithms, which differ by integer multiples of $2\pi i$. A complex logarithm function is a continuous choice of a logarithm on an open subset of $\mathbb{C} \setminus \{0\}$.