linguistic:translation, stylistic:tone, stylistic:style_transfer, linguistic:paraphrasing
Solid angles in arbitrary dimensions

The solid angle subtended by the complete (d − 1)-dimensional spherical surface of the unit sphere in d -dimensional Euclidean space can be defined in any number of dimensions d. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula Ω d = 2 π d 2 Γ (d 2), {\displaystyle \Omega _{d}={\frac {2\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}},} where Γ is the gamma function. When d is an integer, the gamma function can be computed explicitly. It follows that Ω d = { 1 (d 2 − 1) ! 2 π d 2 d even (1 2 (d − 1)) ! (d − 1) ! 2 d π 1 2 (d − 1) d odd. {\displaystyle \Omega _{d}={\begin{cases}{\frac {1}{\left({\frac {d}{2}}-1\right)!}}2\pi ^{\frac {d}{2}}\ &d{\text{ even}}\\{\frac {\left({\frac {1}{2}}\left(d-1\right)\right)!}{(d-1)!}}2^{d}\pi ^{{\frac {1}{2}}(d-1)}\ &d{\text{ odd}}.\end{cases}}}

This gives the expected results of 4 π steradians for the 3D sphere bounded by a surface of area 4π r and 2 π radians for the 2D circle bounded by a circumference of length 2π r. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval and this is bounded by two limiting points.

The counterpart to the vector formula in arbitrary dimension was derived by Aomoto and independently by Ribando. It expresses them as an infinite multivariate Taylor series: Ω = Ω d | det (V) | (4 π) d / 2 ∑ a → ∈ N 0 (d 2) α → a →. {\displaystyle \Omega =\Omega _{d}{\frac {\left|\det(V)\right|}{(4\pi)^{d/2}}}\sum _{{\vec {a}}\in \mathbb {N} _{0}^{\binom {d}{2}}}\left{\vec {\alpha }}^{\vec {a}}.} Given d unit vectors v → i {\displaystyle {\vec {v}}_{i}} defining the angle, let V denote the matrix formed by combining them so the i th column is v → i {\displaystyle {\vec {v}}_{i}}, and α i j = v → i ⋅ v → j = α j i, α i i = 1 {\displaystyle \alpha _{ij}={\vec {v}}_{i}\cdot {\vec {v}}_{j}=\alpha _{ji},\alpha _{ii}=1}. The variables α i j, 1 ≤ i < j ≤ d {\displaystyle \alpha _{ij},1\leq i<j\leq d} form a multivariable α → = (α 12, …, α 1 d, α 23, …, α d − 1, d) ∈ R (d 2) {\displaystyle {\vec {\alpha }}=(\alpha _{12},\dotsc,\alpha _{1d},\alpha _{23},\dotsc,\alpha _{d-1,d})\in \mathbb {R} ^{\binom {d}{2}}}. For a "congruent" integer multiexponent a → = (a 12, …, a 1 d, a 23, …, a d − 1, d) ∈ N 0 (d 2), {\displaystyle {\vec {a}}=(a_{12},\dotsc,a_{1d},a_{23},\dotsc,a_{d-1,d})\in \mathbb {N} _{0}^{\binom {d}{2}},} define α → a → = ∏ α i j a i j {\textstyle {\vec {\alpha }}^{\vec {a}}=\prod \alpha _{ij}^{a_{ij}}}. Note that here N 0 {\displaystyle \mathbb {N} _{0}} = non-negative integers, or natural numbers beginning with 0. The notation α j i {\displaystyle \alpha _{ji}} for j > i {\displaystyle j>i} means the variable α i j {\displaystyle \alpha _{ij}}, similarly for the exponents a j i {\displaystyle a_{ji}}. Hence, the term ∑ m ≠ l a l m {\textstyle \sum _{m\neq l}a_{lm}} means the sum over all terms in a → {\displaystyle {\vec {a}}} in which l appears as either the first or second index.Where this series converges, it converges to the solid angle defined by the vectors.