23 Mar
2011
23 Mar
'11
7:03 p.m.
Can anyone find a closed form expression for the integral of exp[(-1/2) (a[1] x[1]^2 + a[2] x[2]^2 + ... + a[n] x[n]^2)] over the unit (n-1)-sphere? In the n=2 case, we have a x^2 + b y^2 = a cos(t)^2 + b sin(t)^2 = (1/2) ((a+b) + (a-b) cos(2t)), and integrating over t in [0, 2 pi] gives 2 pi exp((-1/4) (a+b)) BesselI(0, (1/4) (a-b)). -- Gene