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 ?
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.
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.
Half car and half plane, the futuristic vehicle allows you to jet off into the sky when you're stuck in traffic.
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.
Schemes I and II have been known to imply the existence of the positive half plane only in physics until now.
The system is stable if all poles have negative real parts, that is, all poles lie in the left half plane.
alpha],[delta]] included in the right half plane {w: Re w > [[[alpha]+[delta]]/[[alpha]+1]]}, where [[DELTA].
The first volume includes a bibliography of Gelbart followed by ten papers on the trace formula, global integrals of Ranking-Selberg type and Shimura type, Gauss sum combinatorics and metaplectic Eisenstein series, a partial Poincare series, restrictions of Saito-Kurokawa representations, models for certain residual representations of unitary groups, crown theory for the upper half plane, unitary periods and Jacquet's relative trace formula, the symmetric powers of cusp forms on GL(2), and the cohomological approach to cuspidal automorphic representations.
As an example of our techniques, we derive the formula for the map from a disk with circular holes to a half plane with radial slits.
In the specific case of a diffracting half plane the numerical results obtained were practically the same as those given by Sommerfeld's rigorous theory.
s] is absolutely convergent on some half plane Rs > [[sigma].
In order to evaluate the integral we choose the contour as real axis and a semi circle of large radius in the upper half plane with necessary cut as the branch points k1,k2.