Another Simplify Idiosyncrasy

*To*: mathgroup at smc.vnet.net*Subject*: [mg26458] Another Simplify Idiosyncrasy*From*: "A. E. Siegman" <siegman at stanford.edu>*Date*: Thu, 21 Dec 2000 01:51:59 -0500 (EST)*Organization*: Stanford University*Sender*: owner-wri-mathgroup at wolfram.com

The following sum (which arises in working with Discrete Fourier Transforms) uc[M_] := (1/M) Sum[ a[n] b[m] Exp[I (n k - m k + k) 2 Pi/M ], {n, 0, M - 1}, {m, 0, M - 1}, {k, 0, M - 1}] should Simplify to the general form a[M] b[0] + a[0] b[1] + a[1] b[2] + . . . + a[M-1] b[M] That's what happens with M = 1, 2, 3, 4, 6, 8, 9 and 12 -- --but with M = 5, 7, 10 and 11 the factors that are equally spaced around the unit circle in the complex plane don't simplify out and one gets pages of terms with factors of (-1)^(n/m). Apparently Mathematica can find these roots for some rational fractions n/m but not others. (Not a complaint, just noting the point; I understand that Simplify'ing is a complex and not always universally successful process.)