Re: What assumptions to use to check for orthogonality
- To: mathgroup at smc.vnet.net
- Subject: [mg112915] Re: What assumptions to use to check for orthogonality
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Tue, 5 Oct 2010 05:38:09 -0400 (EDT)
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
Here is what I get.
In[3]:= r = Integrate[Cos[n*Pi*x]*Cos[m*Pi*x], {x, 0, 1}]
Out[3]//InputForm=
(m*Cos[n*Pi]*Sin[m*Pi] - n*Cos[m*Pi]*Sin[n*Pi])/(m^2*Pi - n^2*Pi)
I am hard pressed to imagine a better result from Integrate. A result
littered with Piecewise to handle a non-generic (indeed, discrete) set
of values would be a really bad idea. And one that would be quite hard
to make work in any general manner.
Also observe that the result is (also) nonzero for m = -n and integral.
One way to see this is to let m approach -n.
In[5]:= Limit[r, m->-n]
Out[5]//InputForm= (2 + Sin[2*n*Pi]/(n*Pi))/4
Regarding the result from Simplify, which is (in some sense) generically
correct, see:
http://forums.wolfram.com/mathgroup/archive/2008/Sep/msg00492.html
Daniel Lichtblau
Wolfram Research