Angular sector ($\{r{e}^{i\theta }:r>0,\alpha <\theta <\beta \},0<\beta -\alpha \le 2\pi$)

The region between two rays with a common vertex, with no upper bound on the modulus. In the displayed convention the vertex and boundary rays are excluded. A holomorphic logarithm with $\alpha <\arg z<\beta$ defines powers $z^p=e^{p\mathrm{Log}z}$ there. For real $p>0$ with $p(\beta -\alpha )\le 2\pi$, the power is a conformal bijection onto the sector $p\alpha <\arg w<p\beta$: it multiplies arguments by $p$ and sends moduli bijectively by $r\mapsto r^p$.