Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers

This page keeps two objects distinct and never silently identifies them.

First, the pointwise principal logarithm of Complex logarithms, the principal logarithm, and principal and multivalued complex powers is defined for every $z\mathrm{\ne }0$ by

$\mathrm{Log}z=\log r+i\theta ,z=r(\cos \theta +i\sin \theta ),-\pi <\theta \le \pi \mathrm{.}$

On the negative real axis the principal polar form uses $\theta =\pi$, so $\mathrm{Log}(-1)=i\pi$. That value is a boundary datum of the slit plane, not the value of a holomorphic branch defined across the cut.

Second, Continuous logarithms and continuous arguments along a contour defines a holomorphic logarithm branch of $z$ on an open set $V$ with $0\mathrm{\notin }V$: a holomorphic $L:V\to \mathbb{C}$ with $\exp (L(z))=z$ for every $z\in V$. On the slit plane $S=\mathbb{C}\setminus \{x\in \mathbb{R}:x\le 0\}$ the principal logarithm is exactly the holomorphic branch normalised by $F(1)=0$, by The principal logarithm is the normalised holomorphic branch on the slit plane; in particular $-\pi <\mathrm{Im}\mathrm{Log}z<\pi$ for every $z\in S$.

Consequently "the principal branch of the logarithm" on this page means the holomorphic function on $S$, while the pointwise principal value on the negative axis is a separate quantity: as $z$ approaches $-1$ from above or below, $\mathrm{Im}\mathrm{Log}z$ tends to $\pi$ or $-\pi$, one-sided limits that assign no value on the cut itself. The branch-defined powers below take a holomorphic branch as input and inherit this discipline.