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Math Help - Fourier

  1. #1
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    Fourier

    Given that f(x)= sinx from 0<x< pi, why is (f- sinx) an even function and hence perform fourier cosin series?

    I thought f(x) is an odd function hencd fourier sine?
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    Given that f(x)= sinx from 0<x< pi, why is (f- sinx) an even function and hence perform fourier cosin series?

    I thought f(x) is an odd function hencd fourier sine?
    Can we have the rest of the question please?

    CB
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  3. #3
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    photo 1.pdfphoto 2.pdf

    here is the question
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  4. #4
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    Quote Originally Posted by alexandrabel90 View Post
    photo 1.pdfphoto 2.pdf

    here is the question
    You question is:

    Show that the Fourier series S(x) of:

    f(x)=\left\{ \begin{array}{ll}0, & -\pi \le x \le 0 \\sin(x), & 0<x\le \pi \end{array} \right.

    is:

    S(x)=\frac{2}{\pi}\left(\frac{1}{2}+\sum_{k=1}^\in  fty \frac{\cos(2\pi x)}{1-4k^2} \right )+\frac{1}{2}\sin(x)

    The answer to your original quaetion is because:

    f(x)=\left|\frac{\sin(x)}{2} \right|+\frac{\sin(x)}{2} \ \ -\pi \le x \le \pi

    The first term on the right is even and so has a Fourier series with cosine terms only and the second is already in Fourier form and indeed is the right most term in the given series.

    (there may be errors in the LaTeX above, but I can't read the rendered images on this machine, but they look OK on CodeCogs)

    CB
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  5. #5
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    how do you know that you need to split the f(x) into one with absolute values and not use f(x)= sin x directly thus considering it as an odd function?
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  6. #6
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    Quote Originally Posted by alexandrabel90 View Post
    how do you know that you need to split the f(x) into one with absolute values and not use f(x)= sin x directly thus considering it as an odd function?
    f(x) is neither odd nor even. A general function on an interval symettric about the origin can always be split into the sum of an even and an odd function (there is even a systematic method of doing so).

    All I have done is split your function into the sum of an odd and even fuction. It is convienient that the odd part is already in the form of a Fourier series and so needs no further work.

    CB
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  7. #7
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    Then in what type of question only then can we say that the function is even or odd and hence only need to compute fourier cos or sine?

    Eg. If a function can be split up, then we cant say that f(x)=x^2 is even right?
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  8. #8
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    Quote Originally Posted by alexandrabel90 View Post
    Then in what type of question only then can we say that the function is even or odd and hence only need to compute fourier cos or sine?

    Eg. If a function can be split up, then we cant say that f(x)=x^2 is even right?
    Can you repost that in a form of English that I can understand please?

    CB
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  9. #9
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    Quote Originally Posted by alexandrabel90 View Post
    how do you know that you need to split the f(x) into one with absolute values and not use f(x)= sin x directly thus considering it as an odd function?
    You don't need to know to split this at all, just compute both sets of coefficients in the usual way and you will get the given result (assuming it is right).

    CB
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