point at infinity


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point at infinity

[′pȯint at in′fin·əd·ē]
(mathematics)
A single point that is adjoined to the complex plane so that it corresponds to the pole of a stereographic projection of the Riemann sphere onto the complex plane, giving the complex plane a compact topology.
References in periodicals archive ?
(35), following the receipt in [27], through the procedure of expanding around the point at infinity described in detail in [39]:
Let O denote the point at infinity on [E.sup.(n)l.sub.[tau]].
The experience of reading the Essais evokes a point at infinity too.
Its capacity to generate movement toward a point at infinity would have suited a king invested in the notion of "the king's two bodies," (20) whereby as "the king" approaches death (at which point one will say "le roi est mort"), "the king" does not approach death (and so one will also say "vive le roi").
By Proposition 6, if Max{([partial derivative]F/[partial derivative]x)(a, b, 0), ([partial derivative]F/[partial derivative]y) (a, b, 0), ([partial derivative]F/[partial derivative]z)(a, b, 0)} < [epsilon], then (a : b : 0) will be a singular point at infinity.
Let [OMEGA] [subset] H be a domain whose extended boundary contains the point at infinity and suppose that there exists a real point t [member of] R [intersection] [OMEGA] such that [OMEGA] \ (-[-[infinity], t] (or [OMEGA] \ [t, + [infinity])) is a slice domain.
Let [OMEGA] [subset] H be a domain whose extended boundary contains the point at infinity and such that, for all I [member of] S, [[OMEGA].sub.I] = [OMEGA] [intersection] [L.sub.I] is simply connected.
As shown in Figure 2, beginning with [sigma] = 0 (the Neumann problem), we will allow [sigma] to move along a ray inclined at an angle [theta] to the horizontal toward the point at infinity. We may then track the trajectory of each Neumann eigenvalue in order to determine whether it eventually approaches a Dirichlet eigenvalue (in which case it will be called an IBC-Dirichlet mode) or migrates to infinity (in which case it will be called a "missing mode").
The Euclidean plane [E.sup.2] = {x = ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [R.sup.3] | [x.sub.3] = 0} and the point at infinity 1 compose the boundary at infinity [partial derivative][H.sup.3] of [H.sup.3].
Let [Q.sub.0] = ([X.sub.0],[Y.sub.0]) be the image on (4) of the point at infinity on (3), then we have [X.sub.0] = 7 + 2[alpha] + 3[[alpha].sup.2] and Y = -17 - 15[alpha] - 8[[alpha].sup.2] where [alpha] = [cube root of (2)].