Is the function $f(w) = e^{-w}w^{z-1}$ holomorphic for $z \in \mathbb{C}$ [closed]

If we choose a branch of the natural log and write $w = e^{\ln(w)}$ then where $\ln(w)$ is continuously defined we can express $f(w) = \exp(-w + \ln(w)(z-1)),$ which is the composition and product of holomorphic functions and hence holomorphic. This can be done on the complement of any closed ray $r e^{i\theta}$ for $\theta$ fixed and $r \geq 0$.

This is not extendable to the entire complex plane unless $z$ is an integer (in which case, choosing different definitions of $\ln(w)$ corresponds to multiplication by $2n\pi i = 1$). For instance, with $z = 1/2$, we have a factor of $\sqrt{w}$, which cannot be defined on the whole complex plane continuously. One should also be careful that our function depends on the choice of branch of $\ln(w)$.