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Math Help - [SOLVED] Fourier series help

  1. #1
    fdddd
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    [SOLVED] Fourier series help

    Let f(x) = sin x for 0 < x < pi . Let a0 + summation from n = 1 to infinity an cosnx be the fourier cosine series which represents f(x) find the value of the coefficient a4.

    when i attempted this question when integrating from -pi to pi f(x) cosnx, i treated f(x) to be 0 from x= -pi to 0 since its not defined. however, in the solution, they extended f(x) to be an even function on -pi to pi. why is it that they can do that?
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  2. #2
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    Quote Originally Posted by fdddd View Post
    Let f(x) = sin x for 0 < x < pi . Let a0 + summation from n = 1 to infinity an cosnx be the fourier cosine series which represents f(x) find the value of the coefficient a4.

    when i attempted this question when integrating from -pi to pi f(x) cosnx, i treated f(x) to be 0 from x= -pi to 0 since its not defined. however, in the solution, they extended f(x) to be an even function on -pi to pi. why is it that they can do that?
    You want to exand \sin x as a cosine series on (0,\pi). Extend \sin x to (-\pi, \pi) in an even manner. So f(-x) = \sin x for x\in (0,\pi). Then this will be a cosine series if exteneded periodically by 2\pi. The coefficients are:
    \pi a_0 = 2\int_0^{\pi}\sin x dx, \ \pi a_n = 2\int_0^{\pi}\sin x \cos nx dx, \ b_n = 0 \mbox{ for }n\geq 1.
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