The principal logarithm is holomorphic off the branch cut

Statement

Let $S = \mathbb{C} \setminus (-\infty, 0]$ be the slit plane, the complement of the standard branch cut. Let $\operatorname{Log}$ be the principal branch of the logarithm. Then $\operatorname{Log}$ is holomorphic on $S$, and $$\operatorname{Log}'(z) = \frac{1}{z}$$ for each $z \in S$.

Remarks

Proof idea: on $S$, $\operatorname{Log}$ is continuous and it is the inverse of the complex exponential on the strip $\{w : -\pi < \operatorname{Im} w < \pi\}$. The inverse function rule gives $\operatorname{Log}'(z) = 1 / e^{\operatorname{Log} z} = 1/z$. $\operatorname{Log}$ is not continuous at the points of $(-\infty, 0)$. For $x < 0$, its values tend to $\ln|x| + i\pi$ from above the axis and to $\ln|x| - i\pi$ from below.

Explanations

L1 human

Away from the cut along the negative real axis, the principal logarithm behaves like the real logarithm: its derivative is $1/z$. On the cut itself, the angle jumps from about $\pi$ to about $-\pi$, so the function is not even continuous there.