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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,

sorry I just skimmed over your post. What about

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[167]= 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[140]= 0
> 
> Assuming[Element[{n, m}, Integers] && n == m, Simplify[r]]
> 
> Out[184]= 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[155]= 1/2
> 
> Limit[Limit[r, n -> 1], m -> 99]
> 
> Out[187]= 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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