Trigonometric Orthogonality Query

  • Thread starter Emphacy
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  • #1
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Hello everyone, I've wandered PF a few times in the past but never thought I'd join, here I am, how exciting.

To keep it short I'm trying to understand the proof behind Fourier Series and can't quite get to grips with basic trigonometric orthogonality.
I understand that sin and cos are naturally orthogonal but I'm having difficulty understanding this:

[tex]
\int_{-\pi}^{\pi}\cos(mx)\cos(nx)\, dx=\pi\delta_{mn} \,\,\,where\,\delta_{mn}=\begin{cases}
&\text{1 if } m=n \\
&\text{0 if } m\neq n
\end{cases}
[/tex]

It makes sense that the area is [itex]\pi[/itex] if [itex]m=n[/itex] because you get this waveform and zero if m is an integer multiple of n because you get this waveform (here the negative areas cancel the positive areas from [itex]-\pi[/itex] to [itex]\pi[/itex])

BUT... If [itex]m\neq n[/itex] AND is sufficiently close e.g (m=2, n=2.1) you get this waveform, in which case the area from [itex]-\pi[/itex] to [itex]\pi[/itex] is clearly not zero.

What gives? Does this only apply if m and n are different by integer multiples?
Thanks for your time! (:
 
Last edited:

Answers and Replies

  • #2
77
9
For both m,n ∈ ℕ, we have the following proof for this orthogonality relation:

Edit : posted it as an attachment because it wouldn't work otherwise.
 

Attachments

Last edited:
  • #4
77
9
I just realized what you did wrong...

Your waveform has (2.1x) and not (2,1x) in the second cosine.

Try changing 2.1x to 2,1x.

Many computing softwares read the input 2.1 as 2*1.

With 2,1x, you'll get the expected result. ;)
 
  • #5
3
0
The software produces the correct waveforms, it actually glitches when using commas (try cos(2,0x)).
The waveform is what I expected, but it just doesn't match up with the mathematics which states it should equal zero.

It only equals zero if you take the limit from [itex]-\infty[/itex] to [itex]\infty[/itex]
 
  • #6
77
9
The integral does not converge if you take the limit from -∞ to ∞. It only converges (to 0) from -nπ to +nπ where n is an integer. The attached visual representation of the integral in my earlier post shows how the area is equal to 0.
 
Last edited:
  • #7
828
2
Yes, m and n must both be integers. Otherwise, as you pointed out, the two functions are not orthogonal. So, not only must m and n but integer differences of each other, they must both be integers.
 

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