Half Plane

half plane

[′haf ¦plān]
The portion of a plane lying on one side of some line in the plane; in particular, all points of the complex plane either above or below the real axis.

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.

References in periodicals archive ?
with four corner points at 2, 1 + i[square root of 3], -1 + i[square root of 3], and -2 lies in the upper half plane.
Contrary to that, the eigenvalues of the power differential droop control stay on the left half plane.
The concept of an acoustic mirror shall be developed resulting in a design which will enable accurate far field sound measurements via a sectioned model with an acoustic reflector on the half plane.
Because of hidden zeros, we at the first by use of MATLAB software, gain hidden zeros after that, gain output zero direction and transmit zeros to second output, after that will replacement Right Half Plane to Left Half Plane, and monitor behavior of system.
with Re [zeta] > 0 and Im [zeta] > 0 is meromorphically continued from the upper half plane of the complex plane to the lower half plane {[zeta] [member of] C : Re [zeta] > 0, Im [zeta] < 0} across the positive real axis where the continuous spectrum of [H.
Stability of a time delay system can be determined by its eigenvalues, which should be located in the open left half plane [2], [12].
1] denote the 'forward' ray contained in the same half plane which contains [K.
In Section 4, we state two conjectures about this generating function, and provide evidence for them by demonstrating that they hold if we only impose on walks a half plane restriction, or no restriction at all.
In this section the PO expression for the half plane geometry will be obtained from the Huygens-Kirchhoff integral with the impedance boundary condition.
Half car and half plane, the futuristic vehicle allows you to jet off into the sky when you're stuck in traffic.
For concreteness the value h is fixed for example in the lower w half plane (LHP).
In this subsection we give an overview of the problems defined in the upper half plane H = {z [member of] C : 0 < Im z} and in right upper quarter plane [Q.