Oleg Zabluda's blog
Thursday, September 14, 2017
 
Say you have (n-1) dimensional sphere.
Say you have (n-1) dimensional sphere. Consider the area of the belt between longitude - 8.1°S and +8.1°N (cos 8.1°=0.99). What is the proportion of the surface area of the belt compared to the total sphere.

The formula is:

1-(2/sqrt(pi))*(Gamma((n+1)/2)/Gamma(n/2))*2F1(1/2, n/2;(n+2)/2;0.99^2)*0.99^n/n

http://www.wolframalpha.com/input/?i=1-(2%2Fsqrt(pi))*(Gamma((n%2B1)%2F2)%2FGamma(n%2F2))*2F1(1%2F2,+n%2F2;(n%2B2)%2F2;0.99%5E2)*0.99%5En%2Fn
http://www.wolframalpha.com/input/?i=1-(2/sqrt(pi))*(Gamma((n%2B1)/2)/Gamma(n/2))*2F1+(1/2,+n/2;(n%2B2)/2;0.99%5E2)*0.99%5En/n

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