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Math Help - Sum of Product of Sines

  1. #1
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    Sum of Product of Sines

    Hello,

    I'm trying to show that \sum _{j=1}^{N-1} \sin \left(\frac{j \pi y}{N}\right)\sin\left(\frac{j \pi k}{N}\right)=0, if y does not equal k (with y and k both being integers). I've tried using the trig identity for the product of sines. That gives \sum _{j=1}^{N-1} \left [\cos\left (\frac{j \pi (y-k)}{N}\right ) - \cos\left (\frac{j \pi (y+k)}{N}\right )\right ], but I'm still stuck there.

    Any ideas?
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Sum of Product of Sines

    Your sum is equal to zero if y<>k and:

    \left\{\left\{\cos \left(\pi  \left(-\frac{k}{N}+k-y\right)\right)\to -\cos \left(\pi  \left(k-\frac{y}{N}+y\right)\right)+\cos \left(\pi  \left(-\frac{k}{N}+k+y\right)\right)+\cos \left(\pi  \left(k+\left(\frac{1}{N}-1\right) y\right)\right)\right\}\right\}
    Last edited by MaxJasper; August 23rd 2012 at 10:28 PM.
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