If $\log:\mathbb{C} \setminus(-\infty,0]\rightarrow \mathbb{C}$ is the principal branch of the complex logarithm, does it then have a primitive on $\{z \in \mathbb{C} : \operatorname{Re}z >0$}?
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2026-05-04 20:03:46.1777925026
Does the complex logarithm have a primitive on $\operatorname{Re}z >0$
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Yes, of course: the region is simply connected and the logarithm is holomorphic over it.
An antiderivative is $f(z)=z\log z-z$.