Results 1 to 6 of 6

Math Help - fourier series

  1. #1
    Member
    Joined
    Jan 2009
    Posts
    83

    fourier series

    g is periodic of period 2pi
    and, for x 2 (−pi, pi] are

    given by the formulae:

    g(x) = 0, -pi < x </= 0
    sin x 0 < x < pi

    I really dont understand how to go about this, since it seems neither to be an even or odd function. Do I need to integrate separately over -pi to 0 and then 0 to pi?

    many thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    g(x) = \begin{cases} 0 & \text{if } -\pi < x \leq 0 \\ \sin x & \text{if } 0 < x < \pi \end{cases}

    You are correct. Since g(x) is neither even nor odd, you have to split your integrals over the appropriate intervals:

    \begin{aligned} a_n & = \frac{1}{\pi} \int_{-\pi}^{\pi} g(x)\cos (nx) \ dx \\ & = \frac{1}{\pi}  \left( \int_{-\pi}^{0} g(x)\cos (nx) \ dx + \int_{0}^{\pi} g(x)\cos (nx) \ dx  \right) \\ & \ \ \vdots \end{aligned}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2009
    Posts
    83
    sorry

    I follow what I have to do but I end up having to integrate sinx sinnx.. is this right? could somebody please go through the methid?

    many thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,366
    Thanks
    41
    Quote Originally Posted by James0502 View Post
    sorry

    I follow what I have to do but I end up having to integrate sinx sinnx.. is this right? could somebody please go through the methid?

    many thanks
    These might help

    \int_0^{\pi} sin x \sin n x\, dx = \left\{ \begin{array}{cc} \frac{\pi}{2} ,&\mbox{ if } n = 1 \\ 0 , & \mbox{ if } n \ne 1 \end{array} \right.

    \int_0^{\pi} sin x \cos n x\, dx = \left\{ \begin{array}{cc} 0 ,&\mbox{ if } n = 1 \\ - \frac{1+\cos n \pi}{n^2-1} , & \mbox{ if } n \ne 1 \end{array} \right.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jan 2009
    Posts
    83
    Could someone please check my working so far..

    I get a[n] = 1/2pi[((-1)^n+1)/(n+1) + ((-1)^(n+1))/(1-n))]

    a[0] = 1/2

    Is this correct? What about when n = 1? I did as above but this meant I could not eliminate anything for being an odd/even function..

    many thanks
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,366
    Thanks
    41
    Quote Originally Posted by James0502 View Post
    Could someone please check my working so far..

    I get a[n] = 1/2pi[((-1)^n+1)/(n+1) + ((-1)^(n+1))/(1-n))]

    a[0] = 1/2

    Is this correct? What about when n = 1? I did as above but this meant I could not eliminate anything for being an odd/even function..

    many thanks
    Here are the first few terms in the Fourier series

    g \approx \frac{1}{\pi} + \frac{1}{2} \sin x - \frac{2}{3 \pi} \cos 2x - \frac{2}{15 \pi} \cos 4 x - \frac{2}{35 \pi} \cos 6x - \frac{2}{63 \pi} \cos 8x

    and a picture to go with it (the hint of a blue curve is the original whereas the red, the approximation).
    Attached Thumbnails Attached Thumbnails fourier series-fourier-g.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Fourier series to calculate an infinite series
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 4th 2010, 01:49 PM
  2. Complex Fourier Series & Full Fourier Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 9th 2009, 05:39 AM
  3. Fourier Series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 15th 2009, 09:30 AM
  4. Fourier series
    Posted in the Calculus Forum
    Replies: 7
    Last Post: December 9th 2008, 09:51 PM
  5. from fourier transform to fourier series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 1st 2008, 06:35 AM

Search Tags


/mathhelpforum @mathhelpforum