 Complex logarithm

For $z=r{e}^{i\theta }\ne 0$, the complex logarithm consists of all values $\log r+i(\theta +2\pi k)$, $k\in \mathbb{Z}$. They give the same exponential because $e^{2\pi ik}=1$. In particular, the logarithms of one are $2\pi ik$, not just zero.

Choosing a single-valued branch of the complex logarithm requires a domain and an argument convention. A holomorphic branch has derivative $1/z$, but no such branch exists on all of $\mathbb{C}\setminus \{0\}$: integrating that derivative once around the origin gives $2\pi i$, whereas a globally defined primitive would give zero. This is why branch geometry matters in contour calculations.