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Fourier series validity

  1. Apr 26, 2014 #1
    1. The problem statement, all variables and given/known data

    Given ∑[itex]^{∞}_{n=1}[/itex] n An sin([itex]\frac{n\pi x}{L}[/itex]) = [itex]\frac{λL}{\pi c}[/itex] σ(x-[itex]\frac{L}{2}[/itex]) + A sin([itex]\frac{\pi x}{2}[/itex]), where L, λ, c, σ and A are known constants, find An.


    2. Relevant equations

    Fourier half-range sine expansion.

    3. The attempt at a solution

    I understand I should expand the RHS as an odd function with period (-L, L) and then compare the coefficients with the LHS, and I do get to correct result. However I didn't understand why I could do so. I mean, originally RHS is NOT a periodic function, that it certainly does not equal the 'constructed' Fourier sine expansion. So how could the coefficients equal since it's actually a different function?
     
  2. jcsd
  3. Apr 26, 2014 #2

    HallsofIvy

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    How do you know you get the "correct result"? Do you mean you get the result given in your text?

    If so then the given answer is NOT for the given function but for a function defined to be [itex]\frac{\lambda L}{\pi c}\sigma(x-\frac{L}{2})+ A sin(\frac{\pi x}{2})[/itex] on (-L, L) and continued "by periodicity" to the rest of the real numbers.
     
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