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Math Help - Integrating fourier series

  1. #1
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    Integrating fourier series

    ok so i had to find the fourier series of cos(ax) with period 2 pi where a is not an integer.

    So i came to the correct answer iam pretty sure, then they asked me to sub in x =pi into the series to get
    i)
    \pi cot(\pi a) -\frac{1}{a } = 2a\sum_{n = 1}^\infty {-1}^{n }/({a}^{2 }- {n}^{2 })\cos (npi)

    \pi cot(\pi a) -\frac{1}{a } = 2a(\frac{1}{{a}^{ 2}  -{1}^{2 } }+ \frac{1}{{a}^{ 2}  -{2}^{2}}     ........)

    ii)now i have to integrate both sides form a =0 to a=theta to obtain

    \frac{\sin \theta \pi }{\theta \pi  }=(1-\frac{{\theta }^{2 } }{{1}^{ 2}.....)  }

    but this is where it gets hard for me,

    integrating the left hand side of i) i get
    \frac{ln(sin(\pi a)}{\pi  } - \frac{lnsin(a\pi )}{\ pi }

    but when calculating from a=0 isnt it undefined?

    I dont even know where to start on the right hand side of the equation so any help with that would be greatly appreciated.
    Last edited by olski1; May 10th 2011 at 04:02 AM.
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  2. #2
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    Quote Originally Posted by olski1 View Post
    ok so i had to find the fourier series of cos(ax) with period 2 pi where a is not an integer.

    So i came to the correct answer iam pretty sure, then they asked me to sub in x =pi into the series to get
    i)
    \pi cot(\pi a) -\frac{1}{a } = 2a\sum_{n = 1}^\infty {-1}^{n }/({a}^{2 }- {n}^{2 })\cos (npi)

    \pi cot(\pi a) -\frac{1}{a } = 2a(\frac{1}{{a}^{ 2} -{1}^{2 } }+ \frac{1}{{a}^{ 2} -{2}^{2}} ........)

    ii)now i have to integrate both sides form a =0 to a=theta to obtain

    \frac{\sin \theta \pi }{\theta \pi }=(1-\frac{{\theta }^{2 } }{{1}^{ 2}.....) }

    but this is where it gets hard for me,

    integrating the left hand side of i) i get
    \frac{ln(sin(\pi a)}{\pi } - \frac{lnsin(a\pi )}{\ pi }

    but when calculating from a=0 isnt it undefined?

    I dont even know where to start on the right hand side of the equation so any help with that would be greatly appreciated.
    Is this part of an assignment that counts towards your final grade?
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  3. #3
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    okay so figured out left hand side of the equation. But how do i go about integrating the right hand side to get
     \int  2a(\frac{1}{{a}^{ 2}  -{1}^{2 } }+ \frac{1}{{a}^{ 2}  -{2}^{2}}     ........)= (1-\frac{\theta ^2}{ 1^2}).....?
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    Is this part of an assignment that counts towards your final grade?
    Yes it is, did not want answers just guidance. But does not matter any more. It took me a while, but I solved it.
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  5. #5
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    Quote Originally Posted by olski1 View Post
    Yes it is, did not want answers just guidance. But does not matter any more. It took me a while, but I solved it.
    Thankyou for this honest reply.
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