For \(0<\beta\le2\pi\) let \(S_\beta\coloneqq\ts{re^{i\theta}\st r>0,\ 0<\theta<\beta}\) be the open sector of opening \(\beta\), so that \(S_\pi=\HH\). Let \(0<\alpha<2\), and on \(S_{2\pi}=\CC\sm[0,\infty)\) define \(z^\alpha\coloneqq r^\alpha e^{i\alpha\theta}\) for \(z=re^{i\theta}\), \(0<\theta<2\pi\). Then \[ F_\alpha\colon\HH\to S_{\alpha\pi},\qquad F_\alpha(z)=z^\alpha, \] is a biholomorphism, with inverse \(w\mapsto w^{1/\alpha}\) defined by the same choice of argument in \((0,2\pi)\). Equivalently, for \(0<\beta<2\pi\) the map \(z\mapsto z^{\pi/\beta}\) is a biholomorphism \(S_\beta\to\HH\).
With the argument taken in \((0,2\pi)\), \(z\mapsto z^\alpha=e^{\alpha(\ln r+i\theta)}\) is holomorphic on \(S_{2\pi}\). It maps \(re^{i\theta}\) with \(0<\theta<\pi\) to \(r^\alpha e^{i\alpha\theta}\) with \(0<\alpha\theta<\alpha\pi<2\pi\); since \(r\mapsto r^\alpha\) is a bijection of \((0,\infty)\) and \(\theta\mapsto\alpha\theta\) is a bijection \((0,\pi)\to(0,\alpha\pi)\), the map is a bijection \(\HH\to S_{\alpha\pi}\) with inverse \(\rho e^{i\varphi}\mapsto\rho^{1/\alpha}e^{i\varphi/\alpha}\), which is also holomorphic. The second statement is the first with \(\alpha=\beta/\pi\), inverted.
The formula \(z^\alpha\) extends continuously to the boundary of \(\HH\). As \(x\) increases from \(0\) to \(\infty\) along the positive real axis, \(x^\alpha\) increases from \(0\) to \(\infty\) along the positive real axis. As \(x\) increases from \(-\infty\) to \(0\) along the negative real axis, \(x^\alpha=\abs{x}^\alpha e^{i\alpha\pi}\) moves from \(\infty\) to \(0\) along the ray \(\ts{te^{i\alpha\pi}\st t>0}\).