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MathGroup Archive 2006

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Re: Assuming non-integer values in Mathematica simplifications

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71094] Re: Assuming non-integer values in Mathematica simplifications
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 8 Nov 2006 06:07:33 -0500 (EST)
  • Organization: The Open University, Milton Keynes, UK
  • References: <eihm5b$ou6$1@smc.vnet.net>

vladimir wrote:
> I just started using Mathematica. I need to simplify the following expressions assuming that w/Pi is not integer (see below). I used the command Element(w/Pi,Rationals] and Element[w/Pi,Reals], but I still get the answer containing If(w/Pi is Integers ...) in many places, making it difficult to extract the answer for non-integer w/Pi. It seems that the simplification commands in Mathematica do not listen to the assumption statements even when such a statement is given within the simplification command. Does anybody know how to tell Mathematice to stop evaluating the integer cases? Thanks in advance.
> 
> Here is my expression:

Surely, what Mathematic cannot do is reading your mind! FullSimplify 
*does* take in account the assumptions you gave; however,  contrary to 
what you seem to believe, these assumptions does *not* -- and have *no* 
reason to -- exclude integer solutions.

In standard mathematics, an integer *is* a rational. The set of rational 
numbers Q is equal to {x = p/q | p and q are integers and q != 0}. 
Therefore, every natural number {1, 2, 3, ... } is a rational, zero is a 
rational, every negative integer is a rational (and also a real, and 
also a complex).

> FullSimplify[(Sum[1, {k, 0, n - 1}]*Sum[
>             Cos[w*k]*Sin[w*k], {k, 0, n - 1}]*Sum[Sin[w*k]*x[k], {k, 
>               0, n - 1}] - Sum[1, {k, 0, n - 1}]*Sum[Cos[w*k]*x[
>           k], {k, 0, n - 1}]*Sum[Sin[w*k]^2, {k, 
>             0, n - 1}] - Sum[Cos[w*k], {k, 0, 
>         n - 1}]*Sum[Sin[w*k], {k, 0, n - 1}]*Sum[Sin[w*k]*x[
>             k], {k, 0, n - 1}] - Sum[Cos[w*k]*Sin[w*k], {k, 0, n - 1}]*
>           Sum[Sin[w*k], {k, 0, n - 1}]*Sum[x[k], {k, 0, n - 1}] + Sum[Cos[w*
>         k], {k, 0, n - 1}]*Sum[x[k], {k, 0, n - 1}]*Sum[Sin[w*k]^2, {k, 0, 
>             n - 1}] + Sum[Cos[w*
>             k]*x[k], {k, 
>               0, n - 1}]*Sum[Sin[w*k], {k, 0, n - 1}]^2)/(-2*Sum[Cos[
>             w*k], {k, 0, n - 1}]*Sum[Sin[w*
>         k], {k, 0, n - 1}]*Sum[Cos[w*k]*Sin[w*k], {k, 0, 
>             n - 1}] + Sum[Sin[w*k], {k, 0,
>            n - 1}]^2*Sum[Cos[w*k]^2, {k, 0, n - 1}] + Sum[Cos[w*k]*
>           Sin[w*k], {k, 0, n - 1}]^2*Sum[1, {k, 0, n - 1}] + Sum[
>         Cos[w*k], {k, 0, n - 1}]^2*
>             Sum[Sin[w*k]^2, {k, 0, n - 1}] - Sum[1, {k, 0, n - 
>             1}]*Sum[Cos[w*k]^2, {k, 0, n - 1}]*Sum[Sin[w*k]^2, {k, 0, n - 
>             1}]), w/Ï? â?? Rationals]

Consequently, the correct assumption is

             Not[w/Pi \[Element] Integers]
Now, compare

FullSimplify[Sum[1, {k, 0, n - 1}]*
  Sum[Cos[w*k]*Sin[w*k], {k, 0, n - 1}]*
  Sum[Sin[w*k]*x[k], {k, 0, n - 1}],
  w/Pi \[Element] Rationals]

that yields

(1/2)*n*If[w/Pi \[Element] Integers,
  (I^((2*(-1 + n)*w)/Pi)*(-1 + n) + Cos[(-1 + n)*w])*
  Sin[(-1 + n)*w], (-Csc[w])*Sin[n*w]*Sin[w - n*w]]*
  Sum[Sin[k*w]*x[k], {k, 0, -1 + n}]

against

FullSimplify[Sum[1, {k, 0, n - 1}]*
  Sum[Cos[w*k]*Sin[w*k], {k, 0, n - 1}]*
  Sum[Sin[w*k]*x[k], {k, 0, n - 1}],
  !w/Pi \[Element] Integers]

that yields

(-(1/2))*n*Csc[w]*Sin[n*w]*Sin[w - n*w]*
  Sum[Sin[k*w]*x[k], {k, 0, -1 + n}]

Regards,
Jean-Marc


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