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Math Help - fourier series question help me pls T_T

  1. #1
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    fourier series question help me pls T_T

    Q1 )find the Fourier coefficients and Fourier series corresponding to the
    periodic functions

    g(x) = { 3 -2<x<0
    .............-3 0<x<2

    Q2 )Expand f(x) = sin x, -π/2 <x <π/2, in a Fourier series if the period is π.

    hope got ppl can help me, thx~~~~~ T_T_T_T_T_T_T
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by helpless View Post
    Q1 )find the Fourier coefficients and Fourier series corresponding to the
    periodic functions

    g(x) = { 3 -2<x<0
    .............-3 0<x<2
    The Fourier Series periodic with period 2L, L>0 is defined to be:

    (a_0)/2 + sum_{n=1 to infty} [a_n cos(n pi x/L) + b_n sin(n pi x/L)]

    where:

    a_n = (1/L) integral_{x=-L to L} f(x) cos(n pi x/L) dx

    b_n = (1/L) integral_{x=-L to L} f(x) sin(n pi x/L) dx

    Now consider:

    g(x) = 3, -2<x<0
    g(x) =-3 0<x<2

    (for technical reasons it does not matter that g(x) is not defined for
    x=3k, k=0, +/-1, ..)

    Now g(x) is odd so all the integrals for the a terms are zero. So lets now
    consider the b terms, here L=2 so:

    b_n = (1/2) integral_{x=-2 to 2} g(x) sin(n pi x/2) dx

    ..... = (1/2) integral_{x=-2 to 0} 3 sin(n pi x/2) dx + (1/2) integral_{x=0 to 2} (-3) sin(n pi x/2) dx

    but sin is antisymetric so:

    b_n = (1/2) integral_{x=-2 to 0} 3 sin(n pi x/2) dx + (1/2) integral_{x=-2 to 0} 3 sin(n pi x/2) dx

    ..... = integral_{x=-2 to 0} 3 sin(n pi x/2) dx = -6/(pi n) - 6/(pi n) cos(n pi)

    So:

    b_n = -12/(pi n) for n odd (that is =1, 3, 5, ..)
    b_n = 0 for n even.

    RonL
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  3. #3
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    the integral_ is wat? thx
    is that copy all in my answer?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by helpless View Post
    the integral_ is wat? thx
    is that copy all in my answer?
    I don't understand what you are asking here.

    RonL
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  5. #5
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    i means wat is integral_ stands for?
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by helpless View Post
    i means wat is integral_ stands for?
    integral_{x=a to b} f(x) dx

    stands for the integral of f(x) with respect to x from x=a to x=b.

    RonL
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