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Solving for variable inside summations

  1. May 24, 2012 #1

    I have an equation:

    [itex]\frac{q-\bar{q}}{w_{i}^{2}+w'_{j}^2}=\sum^{n}_{i=0}\sum^{m}_{j=1}F_{0}\cdot a_{ij}\cdot cos(w_{i}\cdot x)\cdot cos(w'_{j}\cdot y)[/itex]

    I'm trying to solve for [itex]a_{ij}[/itex]

    I have the solution, but I'm not sure how they came up with it. The solution is:

    [itex]a_{ij}=\frac{\int^{s}_{0}\int^{w/2}_{0}q-\bar{q}/F_{0}\cdot cos(w_{i}\cdot x)\cdot cos(w'_{j}\cdot y)\cdot dy\cdot dx}{(w_{i}^{2}+w'_{j}^2)\cdot s \cdot w/8}[/itex]

    Any help would be appreciated. Even if it's just a small push in the right direction.

  2. jcsd
  3. May 24, 2012 #2


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    Science Advisor

    They are using the fact that, if [itex]\{v_i\}[/itex] is an orthonormal basis for an inner product space and [itex]v= \sum a_iv_i[/itex], then [itex]a_i= <v_i, v>[/itex] where < , > is the inner product.
  4. May 30, 2012 #3
    I'll have to read up on that.. thanks a bunch!
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