It's been too long since I last learned complex analysis. Is the following correct??
Let $\mathbb{R}^+:=[0,+\infty)$. Function $z\mapsto \log(z)$, when function values are restricted within the strip $\left\{z\in\mathbb{C}: \text{Im}(z)\in (0,2\pi)\right\}$, is holomorphic on domain $\mathbb{C}\setminus \mathbb{R}^+$.
It appears to me once the entire positive real line is removed, there aren't problematic points within one branch of the logarithm anymore. Correct?? Thanks
Restrict $\exp\colon\mathbb C\to\mathbb C^\times$ to $$\exp\colon\{z\in\mathbb C:0<\mathrm{Im}(z)<2\pi\}\to\mathbb C\backslash\mathbb R_{\ge0}.$$ Since $\exp$ is bijective and holomorphic, by The inverse of a bijective holomorphic function is also holomorphic the inverse, called log, is also a holomorphism.