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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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