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MathGroup Archive 2000

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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.)
>


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