I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.



> 
> What is the vertex angle of a regular $n$-simplex?
> 
> 
> 


Background: For a vertex $v$ in a convex polyhedron $P$, the vertex angle at $v$ is the proportion of the volume that $P$ occupies in a small ball around $v$. In symbols, $$\angle v=\lim\_{\varepsilon\to 0} \frac{|B(v,\varepsilon)\cap P|}{|B(v,\varepsilon)|}.$$ Up to normalization, this definition agrees with the familiar definition of the angles in the plane, or the solid angle in $3$-space.



Angle of a regular simplex
In the paper by John Leech,
"Sphere packings in Higher Space"
*Canadian Journal of Mathematics*, 1964, which you can find at the [Google book links here](https://books.google.com/books?id=ZgwT9xyHZ4gC&pg=PA675&lpg=PA675&dq=%25252522regular+simplex%25252522+solid+angle+at+vertex&source=bl&ots=Yx9JO0VskH&sig=YNEavV1sBX1jNc94ZWSqgwxRZCk&hl=en&ei=JqBGTZ6_CYv4gAeewbXyAQ&sa=X&oi=book_result&ct=result#v=onepage&q=%22regular%20simplex%22%20solid%20angle%20at%20vertex&f=false),
the following formulas are given for the "solid angular content at each vertex of a regular simplex":


$$2^{-n} n! f\_n(n) H\_n$$


where


$$H\_n = 2 \pi^{\frac{1}{2}n} / \Gamma(n/2)$$


is the total $(n{-}1)$-dimensional surface content,
$f\_n(\sec 2 \alpha)=F\_n(\alpha)$, and (finally!)
$F\_n(\alpha)$ is Schläfli's function mentioned by BS in his comment.
This function is discussed in Section 7.2 (p.107ff) of
Chuanming Zong's book, *Sphere Packings* ([Google books link here](https://books.google.com/books?id=ScdN96_Ux-oC&printsec=frontcover&dq=zong+sphere+packings&source=bl&ots=l5g1c8RkZi&sig=cspsXdNQYQzLiaV5wGsbxRhOBcI&hl=en&ei=5qxGTfX0CoucgQfLutyYAg&sa=X&oi=book_result&ct=result#v=onepage&q&f=false))
and in
the paper, "Analytic structure of Schläfli function,"
Kazuhiko Aomoto,
[*Nagoya Math. J.* Volume 68 (1977), 1-16](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-68/issue-none/Analytic-structure-of-Schl%C3%A4fli-function/nmj/1118796538.full),
also mentioned by BS.
Rogers computed an asymptotic formula for $F\_n(\alpha)$:


$$ \frac{ \sqrt{ 1+cn}} {\sqrt{2} e^{1/c} n!} \left( \frac{2 e}{\pi c n} \right) ^{n/2} \;,$$


where
$c = (\sec 2 \alpha - (n-1))^{-1}$.


