Complex powers defined from a holomorphic logarithm branch

Let $V\subseteq \mathbb{C}$ be open with $0\mathrm{\notin }V$, and let $L:V\to \mathbb{C}$ be a holomorphic logarithm branch of $z$ on $V$: $L$ is holomorphic and $\exp (L(z))=z$ for every $z\in V$, the branch vocabulary being that of the dictionary in Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers. For $\alpha \in \mathbb{C}$ define the branch power

$z_L^{\alpha }:=\exp (\alpha L(z))(z\in V)\mathrm{.}$

The subscript records the branch: the same base $z$ can carry many holomorphic logarithm branches, and different branches give different values in general. The defining expression is well formed because the complex exponential is a total function on $\mathbb{C}$, whose values and addition law are those of $\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential.

The principal branch power is the special case on the slit plane $S=\mathbb{C}\setminus \{x\in \mathbb{R}:x\le 0\}$:

$z_{\mathrm{Log}}^{}:=\exp (\alpha \mathrm{Log}z)(z\in S),$

where $\mathrm{Log}$ is the holomorphic principal branch of the dictionary remark. On $S$ this agrees with the pointwise principal power of the published principal-logarithm definition; on the negative axis the pointwise principal value is still defined while the holomorphic branch power is not, and the two are not silently identified.