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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! (:

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! (:

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