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Math Help - Fourier series

  1. #1
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    Fourier series

    Okay, I dont even know how to solve this ... plzz someone help ^^


    Let f(x) be a periodic function with period P = 2pie defined in one period by

    f(x) = pie if -pie < x < 0,
    pie - x if 0 less than equal x less than equal pie


    (a) Compute An for n greater than and equal to 0.
    (b) Compute Bn for n greater than and equal to 1.
    (c) Now write f(x) as a Fourier series.

    I know i am asking a lot but these type of problems will be on my test -_-
    Thanks
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  2. #2
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    Quote Originally Posted by darkangel View Post
    Okay, I dont even know how to solve this ... plzz someone help ^^


    Let f(x) be a periodic function with period P = 2pie defined in one period by

    f(x) = pie if -pie < x < 0,
    pie - x if 0 less than equal x less than equal pie


    (a) Compute An for n greater than and equal to 0.
    (b) Compute Bn for n greater than and equal to 1.
    (c) Now write f(x) as a Fourier series.

    I know i am asking a lot but these type of problems will be on my test -_-
    Thanks
    From the definition of a Fourier series for a function on the interval (-\pi,\pi), or periodic with period 2\pi :

    a_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx)\ dx, n=0,1,\ ...

    b_n=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin(nx)\ dx, n=1,\ ...

    So what problems are you having with the integrals?

    CB
    Last edited by CaptainBlack; December 9th 2008 at 08:36 PM.
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  3. #3
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    CaptainBlack, the problem is that our professor did not even explain us - how to solve these problems and now he is giving these on the test.

    I have looked many examples, on this board. But the truth is that I dont even know how to solve these.
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  4. #4
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    Quote Originally Posted by darkangel View Post
    CaptainBlack, the problem is that our professor did not even explain us - how to solve these problems and now he is giving these on the test.

    I have looked many examples, on this board. But the truth is that I dont even know how to solve these.
    The previous post gave the definition of the Fourier coeficients, to even be looking at Fourier series you must have covered integration, so you should be able to do the integrals. Further information can be found here.

    But the Fourier series representation of your function is:

    f(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n\cos(nx) + b_n \sin(nx)]


    CB
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  5. #5
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    thanks captainBlack, I appreciate ur help
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  6. #6
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    I have this problem as well.

    The Wiki says that in order to do this problem, f(x) must be integrable from -pi to pi. This step function is not integrable? It's not a continuous function. I guess I could find the integral using geometric sums, but how would I plug that into the Fourier coefficients?
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  7. #7
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    A function doesn't have to be continuous to be integralble.

    f(x) = \begin{cases} {\color{red}\pi} & {\color{red}\text{if } -\pi < x < 0} \\ {\color{blue}\pi - x} & {\color{blue}\text{if } 0 \leq x \leq \pi } \end{cases}

    f(x) is neither even nor odd so split your integral accordingly:
    \begin{aligned} a_n & =\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx)\ dx \\ & = \frac{1}{\pi} \left( \int_{-\pi}^{0} {\color{red}f(x)} \cos (nx) dx \ + \ \int_0^{\pi} {\color{blue}f(x)} \cos (nx) dx\right) \end{aligned}
    \begin{aligned}{\color{white}a_n} & = \frac{1}{\pi} \left( \int_{-\pi}^0 {\color{red}\pi} \cos (nx) \ dx +  \int_0^{\pi}{\color{blue} (\pi - x)} \cos (nx) dx\right) \end{aligned}
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  8. #8
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    Ah, of course! Thank you!
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