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What assumptions to use to check for orthogonality under integration?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112868] What assumptions to use to check for orthogonality under integration?
  • From: "Nasser M. Abbasi" <nma at 12000.org>
  • Date: Mon, 4 Oct 2010 06:06:03 -0400 (EDT)
  • Reply-to: nma at 12000.org

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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