# Half-plane

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The set of points in a plane situated to one side of a given straight line in that plane. The coordinates of the points of a half-plane satisfy an inequality $Ax + By + C > 0$, where $A , B , C$ are certain constants such that $A$ and $B$ do not vanish simultaneously. If the straight line $Ax + By + C = 0$ itself (the boundary of the half-plane) belongs to the half-plane, the latter is said to be closed. Special half-planes on the complex plane $z = x + iy$ are the upper half-plane $y = \mathop{\rm Im} z > 0$, the lower half-plane $y = \mathop{\rm Im} z < 0$, the left half-plane $x = \mathop{\rm Re} z < 0$, the right half-plane $x = \mathop{\rm Re} z > 0$, etc. The upper half-plane of the complex $z$- plane can be mapped conformally (cf. Conformal mapping) onto the disc $| w | < 1$ by the Möbius transformation
$$w = e ^ {i \theta } \frac{z - \beta }{z - \overline \beta \; } ,$$
where $\theta$ is an arbitrary real number and $\mathop{\rm Im} \beta > 0$.