Principal branch of the logarithm and of $z^\alpha$

Let \(\Omega\coloneqq\CC\setminus(-\infty,0]\). Every \(z\in\Omega\) can be written uniquely as \(z=re^{i\theta}\) with \(r>0\) and \(\theta\in(-\pi,\pi)\). The principal branch of the logarithm is the function \(\Log\colon\Omega\to\CC\) given by \[ \Log(z)\coloneqq\log r+i\theta, \] where \(\log r\) is the real natural logarithm. For \(\alpha\in\CC\), the principal branch of \(z^\alpha\) is the function on \(\Omega\) given by \[ z^{\alpha}\coloneqq e^{\alpha\Log(z)}. \]