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Understanding Fourier Coefficients using PDE

  1. Jan 24, 2016 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    In my PDE course we have a homework question stating the following:

    Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients.

    2. Relevant equations

    From my notes on this type of question:

    a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx]

    a_n = c_n + c_(-n) = 1/pi * integral from -pi to pi [f(x) cos(n*x) dx ]

    b_n = i(c_n - c_(-n)) = 1/pi * integral from -pi to pi [f(x) sin(n*x) dx]

    3. The attempt at a solution

    Is it as simple as just a plug and chug based off my noes?

    a_o's integration with f(x) = x just is x^2/(2*pi) from -pi to pi so we have
    a_o = pi/2 - pi/2 = 0

    a_n's integration is just equal to 0 as well.

    b_n is just -2(-1)^n/n

    So thus, the fourier coefficients here are b_n = [(-2)(-1)^n]/n
    for n ≥ 1

    Am I understanding the question properly?
     
  2. jcsd
  3. Jan 24, 2016 #2
    Graph the sum of the first 10 terms and compare it to the original function.
     
  4. Jan 24, 2016 #3

    RJLiberator

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    Since f(x) = x is odd, we can safely say a_n is equal to 0 for n >= 0.

    So, it seems I am right that b_n is the only coefficients.

    The fourier expansion is thus
    2sin(x)- sin(2x) + (2/3)sin(3x)/3 +...


    So I guess my understanding on this problem seems to be getting better.
    My question is thus, according to the question in the initial post, is a complete and safe way to answer this question by stating that the function is odd so a_n coefficients are 0, and so we observe
    b_n = (2/n)(-1)^(n+1)

    I just want to make sure I am answering this question with completeness.
     
  5. Jan 24, 2016 #4
    I'd be happy with your answer if I was grading it. But I'd be happier with a graph showing the series agrees with the initial function.

    My classes always have a heavy emphasis on assessment: showing your answer is right with a method independent of the method originally used to compute the answer.
     
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