X is the unit n-cube, [0, 1]n. Let kn = ∫X cos2( π(x1 + x2 + ... + xn)/(2n) ) dx1 ... dxn. What is limn→∞ kn ?
Solution
Let yi = 1 - xi and change variables from xi to yi. The sum (x1 + x2 + ... + xn)/(2n) becomes 1/2 - (y1 + y2 + ... + yn)/(2n), so the integrand becomes sin2( π(y1 + y2 + ... + yn)/(2n) ). So the integral is identical to the original except that cos has been changed to sin. Thus we can add it to the original to get 2kn = ∫X dx1 dx2 ... dxn = 1. So kn = 1/2 for all n.
© John Scholes
jscholes@kalva.demon.co.uk
25 Jan 2002