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>

> 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:

*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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