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Given two functions $f$ and $g$ which both have period $2\pi$ and average $0$ over the period, that is $f(x)=f(x+2\pi)$ and $\int_0^{2\pi} f(x) \rm{dx}=0$ (and the same for $g$) and also $f$ is known to be odd (or even, a shift in $x$ means it can be either). What can be said about the integral of their product over the period? $$\int_0^{2\pi} f(x)g(x)\rm{dx}=?$$ I'm hoping for this to be zero, so also - what relationships between the $f$ and $g$ would make this zero? ($f$ and $g$ can be assumed to be infinitely differentiable, finite, etc.)

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  • $\begingroup$ The integral doesn't need to be $0$. Take for example $g = \pm f$. $\endgroup$
    – dxiv
    Oct 10, 2016 at 15:44
  • $\begingroup$ I realise it isn't necessarily $0$. But I'm looking for relations which would make it $0$. Or just more general statements about the integral. $\endgroup$
    – Kvothe
    Oct 10, 2016 at 15:47
  • $\begingroup$ There is no general necessary condition other than the statement itself that $f \cdot g$ must have $0$ integral over the period. You could find sufficient conditions for this to hold, but it's not clear what kind of relations you are looking for. For example, the integral is $0$ if $f$ is differentiable and $g = f'$. $\endgroup$
    – dxiv
    Oct 10, 2016 at 15:59
  • $\begingroup$ Sorry, I don't know what kind of relations I'm looking for either - I'm just trying to find any set of conditions on $g$ for the integral to be zero so I can possibly go back to some previous work and see if those conditions would be physical. $\endgroup$
    – Kvothe
    Oct 10, 2016 at 17:54
  • $\begingroup$ $g=\lambda f' + \mu$ is one family of such $g$. But there are infinitely many others. $\endgroup$
    – dxiv
    Oct 10, 2016 at 18:16

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