# Half Plane

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## half plane

[′haf ¦plān]## Half Plane

in mathematics, the set of points of a plane that lie on one side of some line in the plane. The coordinates of points in a half plane satisfy the inequality *Ax* + *By* + *C* > 0, where *A, B*, and *C* are constants such that *A* and *B* are not simultaneously equal to zero. The line *Ax + By + C* = 0 is called the boundary of the half plane. If the boundary is included in the half plane, the half plane is said to be closed.

The complex plane *z* = *x* + *iy* contains an upper half plane *y* = Im *z* > 0, a lower half plane *y* = Im *z* < 0, a left half plane *x* = Re *z* < 0, a right half plane *x* = Re *z* > 0, and so on. The upper half plane of the complex *z*-plane is mapped conformally onto the circle ǀwǀ < 1 by the linear fractional function

where *θ* is an arbitrary real number and Im *β* > 0.