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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
In particular, one can describe explicitly the hyperbolic geometry of such surfaces and one can compute scalar field trajectories on ([summation], G) by determining trajectories of an appropriate lift of the model to the Poincare half plane H and projecting them to [D.sup.*] or to A(R) through the uniformization map.
Mochizuki [10] introduced the Nevanlinna class [N.sub.0](D) and the Smirnov class [N.sub.*](D) on the upper half plane D ??= {z [member of] C | Im z >0}: the class [N.sub.0](D) is the set of all holomorphic functions f on D satisfying
Suppose that the function q [member of] [A.sup.*] is a univalent mapping of U into the right half plane with q(0) = 1 and
Xu, "Solution for a circular cavity in an elastic half plane under gravity and arbitrary lateral stress," International Journal of Rock Mechanics and Mining Sciences, vol.
[16] studied the scattering of plane waves by a cylindrical cavity with lining in a poroelastic half plane using the complex variable function method.
Falope investigate the contact problem of an Euler-Bernoulli nanobeam of finite length bonded to a homogeneous elastic half plane. The analysis is performed under plane strain condition.
For example, Huang and Yu [11] studied an elastic half plane under surface loading with consideration of surface energy effects.
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.sub.d] is located.
Stability of a time delay system can be determined by its eigenvalues, which should be located in the open left half plane [2], [12].
In each iterative process, the according region in Figure 1 is used if all eigenvalues defined as [LAMBDA] = {[[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.n]] lie in the left half plane. In this case, the corresponding region [[OMEGA].sub.[LAMBDA]] = [OMEGA]([[alpha].sub.[LAMBDA]], [[xi].sub.[LAMBDA]], [[omega].sub.[LAMBDA]]) determined by [LAMBDA] = {[[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.n]] will be compared with the respected goal region.
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.