       Re: What assumptions to use to check for orthogonality

• To: mathgroup at smc.vnet.net
• Subject: [mg112904] Re: What assumptions to use to check for orthogonality
• From: Patrick Scheibe <pscheibe at trm.uni-leipzig.de>
• Date: Tue, 5 Oct 2010 05:35:54 -0400 (EDT)

```Hi,

Clear[n, m, x]
r = Integrate[Cos[n*x]*Cos[m*x], {x, -Pi, Pi}]
(* The case m=n *)
FullSimplify[Limit[r, m -> n], Element[n, Integers]]
(* The case m!=n *)
FullSimplify[r,
Element[{n, m}, Integers] && n != m]

Cheers
Patrick

On Mon, 2010-10-04 at 06:06 -0400, Nasser M. Abbasi wrote:
> This is basic thing, and I remember doing this or reading about it before.
>
> I am trying to show that Cos[m Pi x], Cos[n Pi x] are orthogonal
> functions, m,n are integers, i.e. using the inner product definition
>
> Integrate[Cos[n*Pi*x]*Cos[m*Pi*x], {x, 0, 1}];
>
> So, the above is ZERO when n not equal to m and 1/2 when n=m. hence
> orthogonal functions.
>
> This is what I tried:
>
> ------ case 1 -------------
> Clear[n, m, x]
> r = Integrate[Cos[n*Pi*x]*Cos[m*Pi*x], {x, 0, 1}];
> Assuming[Element[{n, m}, Integers], Simplify[r]]
>
> Out= 0
> ----------------
>
> I was expecting to get a result with conditional on it using Piecewise
> notation.
>
> Then I tried
>
> ---------case 2 ------------
> Clear[n, m, x]
> r = Integrate[Cos[n*Pi*x]*Cos[m*Pi*x], {x, 0, 1}];
> Assuming[Element[{n, m}, Integers] && n != m, Simplify[r]]
>
> Out= 0
>
> Assuming[Element[{n, m}, Integers] && n == m, Simplify[r]]
>
> Out= Indeterminate
> ----------------
>
> So, it looks like one has to do the limit by 'hand' to see that for n=m
> we get non-zero?
>
> -------------------
> Clear[n, m, x]
> r = Integrate[Cos[n*Pi*x]*Cos[m*Pi*x], {x, 0, 1}];
> Limit[Limit[r, n -> m], m -> 1]
>
> Out= 1/2
>
> Limit[Limit[r, n -> 1], m -> 99]
>
> Out= 0
> ----------------------------
>
>
> So, is there a way to get Mathematica to tell me that the integral is
> zero for m!=n and 1/2 when n=m? (tried Reduce, Refine). It seems the
> problem is that the Integrate is not taking the limit automatically to
> determine what happens when n=m? Should it at least in case have told me
> that when n!=m it is zero, and when n=m it is  Indeterminate? It just
> said zero which is not correct when n=m and I did say n,m are integers.
>
> thanks
> --Nasser
>

```

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