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Math Help - Half Range Fourier Series

  1. #1
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    Half Range Fourier Series

    If f(x) is defined on [0,L] then it can be extended to be either odd or even giving formulas
     f(x)=\sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L} and
    f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac  {\pi nx}{L} respectively
    If f(x) was not even or odd, is there another formula that can be used to express it?
    I'm trying to get something of the form
    f(x)=\sum_{n=0}^{\infty}c_n\sin\frac{(2n+1)\pi x}{2L}, where c_n is some constant expressed as an integral of f, but I've no idea how to show this.

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    Last edited by Rachel123; January 29th 2013 at 11:59 AM.
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  2. #2
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    Re: Half Range Fourier Series

    The Fourier basis functions \sin and \cos are eigenvectors of the operator \nabla^2. Recall that eigenvectors satisfy \nabla^2 u = -\lambda^2 u. Then u = a\cos(\lambda x)+b\sin(\lambda x). If you have (zero) Dirichlet boundary conditions, you get eigenvectors u = \sin(\lambda x) where \lambda = {\pi n\over L}. If you have (zero) Neumann boundary condition u = \cos({\pi n\over L} x) are eigenvectors. You can also have periodic boundary conditions, which yields the traditional Fourier series.
    Now you ask about u = \sin\left({\pi (2n+1)\over 2L} x\right) eigenvectors. I think you can make it work with u(0)=0 and {du\over dx}(L)=0.
    Last edited by vincisonfire; January 29th 2013 at 06:57 PM.
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  3. #3
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    Re: Half Range Fourier Series

    I don't understand this, we have not yet covered eigenvectors relating to fourier series and Dirichlet boundary condition yet. Is there a simpler way to show this?
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  4. #4
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    Re: Half Range Fourier Series

    I think you can just break it up into odd and even parts - so f(x) = f_{odd}(x)+f_{even}(x)=

    \sum_{n=1}^{\infty}b_n\sin\frac{\pi nx}{L} +\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}=

    \frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos\frac {\pi nx}{L}+b_n\sin\frac{\pi nx}{L}

    And by the way, f_{odd}(x)=\frac{f(x)-f(-x)}{2} and f_{even}(x)=\frac{f(x)+f(-x)}{2}.

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