English Dictionary

Riemann sphere

名詞

(topology, complex analysis) The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.

日本語の意味

複素数全体に∞を加えたもの。複素平面に無限遠点を追加することで得られる数学的対象であり、3次元ユークリッド空間の球面と同相になる。 / トポロジーや複素解析における概念。複素数体を一つの理想的な無限遠点によって拡張したもの。

When mapping meromorphic functions to the Riemann sphere, poles correspond to the added point at infinity.

メロモルフィック関数を無限遠点を加えた拡大複素平面（無限遠点を含む複素数全体）へ写すと、極は追加された無限遠点に対応します。

意味（1）

(topology, complex analysis) The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.

We use ̂Complex⨯ (resp. Complex ) to denote the subset z∈̂ Complex :z ne 0 (resp. z∈̂ Complex :z ne ∞) in the Riemann sphere ̂ Complex .

意味（2）

(topology, complex analysis) The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.

Every circle 𝛾 on the Riemann sphere 𝛴 which does not go through a given point P*∈𝛴 divides 𝛴 into two parts, such that one part contains P* and the other does not.

Riemann spheres （ plural ）