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Re: Assuming non-integer values in Mathematica simplifications

• To: mathgroup at smc.vnet.net
• Subject: [mg71018] Re: [mg71002] Assuming non-integer values in Mathematica simplifications
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 4 Nov 2006 23:07:16 -0500 (EST)
• References: <200611040908.EAA25119@smc.vnet.net>

On 4 Nov 2006, at 18:08, 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:
>
> 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]
>

Of course using Element(w/Pi,Rationals] etc would not do, since  
integers are rationals etc.

You need to use this double assumption in FullSimplify:

Not[Element[w/(2 Pi), Integers]] && Not[Element[w/Pi , Integers]]

Of course this is logically equivalent to simply
Not[Element[w/Pi , Integers]]
but Mathematica can't make this sort of reduction. Note also that  
even this remains unsimplified:

FullSimplify[Not[Element[a/2, Integers] && Not[Element[a, Integers]]]]

a/2 \[NotElement] Integers && a \[NotElement] Integers


Andrzej Kozlowski
Tokyo, Japan

• References:
  • Assuming non-integer values in Mathematica simplifications
    • From: vladimir <gpwr9k95@yahoo.com>