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Math Help - Fourier series cosine

  1. #1
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    Fourier series cosine

    Hello can anybody help me with this problem?
    Find the even (cosine) extension of the function given in
    Q. 6 as a Fourier-series. Write down, without making any calculations, the odd extension as a Fourier

    This is the fuction given in question 6

    f
    (x) = sin (x)
    0 (less than or equal to) x (less than or equal to) (pi);
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  2. #2
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    Quote Originally Posted by math_lete View Post
    Hello can anybody help me with this problem?








    Find the even (cosine) extension of the function given in
    Q. 6 as a Fourier-series. Write down, without making any calculations, the odd extension as a Fourier

    This is the fuction given in question 6


    f
    (x) = sin (x)


    0 (less than or equal to) x (less than or equal to) (pi);

    <br />
y_c = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos n x<br />
where a_n = \frac{2}{\pi}\int_0^{\pi} \sin x \cos n x\,dx

    For the fourier sine series there's just one term

    Spoiler:
    y_s = \sin x
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  3. #3
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    Quote Originally Posted by math_lete View Post
    Hello can anybody help me with this problem?
    Find the even (cosine) extension of the function given in
    Q. 6 as a Fourier-series. Write down, without making any calculations, the odd extension as a Fourier

    This is the fuction given in question 6

    f
    (x) = sin (x)
    0 (less than or equal to) x (less than or equal to) (pi);
    The "even extension would be, of course, f(x)= -sin(x) for -\pi\le x\le 0, sin(x) for [tex]0\le x\le \pi[tex]. Apply the usual formulas for Fourier coefficients to that. Simplifications: because this is an even function, the sine coefficients will be 0 and you can get the cosine coefficients by integrating from 0 to \pi and doubling.

    The "odd extension" which, as implied, you can do without computation, is just sin(x) for -\pi\le x \le \pi and the Fourier series is just sin(x) itself- that is all coefficients of cos(x) are 0, the coefficient of sin(nx) is 0 unless n= 1 in which case it is 1.
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