Half Plane

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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 ?
Coman likes to hog the touchline, opening room for the attacking full-back or the half-space channels for a midfielder.
Transient loading of elastic half-space by a point force, whose time distribution is given by the Heaviside function
The closed form solution for stresses due to uniform cuboidal eigenstrains in an elastic isotropic infinite space, as well as in the half-space, was advanced by Chiu [7] and by Chiu [8] respectively, using the mirror image method.
The subsoil has been introduced into the calculations by means of three basic subsoil models: half-space model, two-parametric model and one-parametric (Winkler) subsoil model.
The first part of the monograph finds representations for the best constants in the Miranda-Agmon maximum principle for solutions of homogeneous strongly elliptic systems of the second order with constant coefficients in a half-space, then turns to pointwise estimates for derivatives of solutions of elliptic equations.
Stokes' first problem for a Rivlin-Ericksen fluid of second grade in a porous half-space was developed by Jordan and Puri [8].
We consider a thermo-elastic half space with diffusion with free surface along x-axis and negative z-axis is taken downward into the half-space as shown in Fig 1.
In this case, due to a significant z-direction conductivity increase, the initial half-space may be approximated with a high conductivity plate of finite thickness.
Georgia) investigates the boundary properties of the differentiated Poisson integral for various domains such as circle, ball, half-plane, half-space, and bicylinder.
Based on the changes in slope the power spectrum curve for the area shows that there are three major density layers overlying a half-space.