Re: Another Simplify Idiosyncrasy

*To*: mathgroup at smc.vnet.net*Subject*: [mg26470] Re: [mg26458] Another Simplify Idiosyncrasy*From*: BobHanlon at aol.com*Date*: Fri, 22 Dec 2000 22:46:09 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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}] ; If you have lots of time to wait, you can use FullSimplify FullSimplify[uc[5]] a[4]*b[0] + a[0]*b[1] + a[1]*b[2] + a[2]*b[3] + a[3]*b[4] FullSimplify[uc[7]] a[6]*b[0] + a[0]*b[1] + a[1]*b[2] + a[2]*b[3] + a[3]*b[4] + a[4]*b[5] + a[5]*b[6] Note that your expression for the simplified result should read: a[M - 1] b[0] + a[0] b[1] + a[1] b[2] + ...+a[M - 2] b[M - 1] Bob Hanlon In a message dated 12/21/00 2:46:24 AM, siegman at stanford.edu writes: >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.) >