Editing Solid angle (section)

== Solid angles in arbitrary dimensions ==
The solid angle subtended by the complete ({{mvar|d − 1}})-dimensional spherical surface of the unit sphere in [[Euclidean space|{{math|''d''}}-dimensional Euclidean space]] can be defined in any number of dimensions {{math|''d''}}. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula
<math display="block">\Omega_{d} = \frac{2\pi^\frac{d}{2}}{\Gamma\left(\frac{d}{2}\right)}, </math>
where {{math|Γ}} is the [[gamma function]]. When {{math|''d''}} is an integer, the gamma function can be computed explicitly.<ref>{{cite journal| last = Jackson| first = FM| year = 1993| title = Polytopes in Euclidean n-space| journal = Bulletin of the Institute of Mathematics and Its Applications| volume = 29| issue = 11/12| pages = 172–174| url = https://www.researchgate.net/publication/265585180}}</ref> It follows that
<math display="block">
  \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}
</math>

This gives the expected results of 4{{pi}} steradians for the 3D sphere bounded by a surface of area {{math|4π''r''<sup>2</sup>}} and 2{{pi}} radians for the 2D circle bounded by a circumference of length {{math|2π''r''}}. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered  1D "sphere" is the interval {{closed-closed|−''r'', ''r''}} and this is bounded by two limiting points.

The counterpart to the vector formula in arbitrary dimension was derived by Aomoto<ref>{{cite journal|first=Kazuhiko| last=Aomoto| title=Analytic structure of Schläfli function| journal=Nagoya Math. J.| volume=68| year=1977| pages=1–16| doi=10.1017/s0027763000017839| doi-access=free}}</ref><ref>{{cite journal| last1=Beck|first1=M.|last2=Robins|first2=S.|last3=Sam|first3=S. V. |year=2010 |title=Positivity theorems for solid-angle polynomials |journal=Contributions to Algebra and Geometry |volume=51|issue=2| pages=493–507 |arxiv=0906.4031 |bibcode=2009arXiv0906.4031B}}</ref>
and independently by Ribando.<ref>{{cite journal| journal=Discrete & Computational Geometry| volume=36| issue=3| pages=479–487| year=2006| title= Measuring Solid Angles Beyond Dimension Three| first=Jason M.| last=Ribando| doi=10.1007/s00454-006-1253-4| doi-access=free}}</ref> It expresses them as an infinite multivariate [[Taylor series]]:
<math display="block">\Omega = \Omega_d \frac{\left|\det(V)\right|}{(4\pi)^{d/2}} \sum_{\vec a\in \N_0^{\binom {d}{2}}}
   \left [
     \frac{(-2)^{\sum_{i<j} a_{ij}}}{\prod_{i<j} a_{ij}!}\prod_i \Gamma \left (\frac{1+\sum_{m\neq i} a_{im}}{2} \right )
    \right ] \vec \alpha^{\vec a}.
 </math>
Given {{mvar|d}} unit vectors <math>\vec{v}_i</math> defining the angle, let {{mvar|V}} denote the matrix formed by combining them so the {{mvar|i}}th column is <math>\vec{v}_i</math>, and <math>\alpha_{ij} = \vec{v}_i\cdot\vec{v}_j = \alpha_{ji}, \alpha_{ii}=1</math>. The variables <math>\alpha_{ij},1 \le i < j \le d</math> form a multivariable <math>\vec \alpha = (\alpha_{12},\dotsc , \alpha_{1d}, \alpha_{23}, \dotsc, \alpha_{d-1,d}) \in \R^{\binom{d}{2}}</math>. For a "congruent" integer multiexponent <math>\vec a=(a_{12}, \dotsc, a_{1d}, a_{23}, \dotsc , a_{d-1,d}) \in \N_0^{\binom{d}{2}}, </math> define <math display="inline">\vec \alpha^{\vec a}=\prod \alpha_{ij}^{a_{ij}}</math>. Note that here <math>\N_0</math> = non-negative integers, or natural numbers beginning with 0. The notation <math>\alpha_{ji}</math> for <math>j > i</math> means the variable <math>\alpha_{ij}</math>, similarly for the exponents <math>a_{ji}</math>. 
Hence, the term <math display="inline">\sum_{m \ne l} a_{lm}</math> means the sum over all terms in <math>\vec a</math> 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.