Right half-plane ($\{z\in \mathbb{C}:\mathrm{Re}z>0\}$)

The right half-plane consists of complex numbers with positive real part. Its boundary is the imaginary axis. For example, $1+i$ is inside, $i$ is on the boundary, and $-1+i$ is outside. Rotation $z\mapsto iz$ takes the right half-plane onto the upper half-plane; the Cayley transform between the half-plane and disk uses $R(z)=(z-1)/(z+1)$ for this convention.