Results 1 to 3 of 3
Like Tree3Thanks
  • 3 Post By romsek

Thread: How to integrate from a Fourier sine coefficient?

  1. #1
    Newbie
    Joined
    Jan 2019
    From
    UK
    Posts
    5

    Question How to integrate from a Fourier sine coefficient?

    I try to solve a Fourier series (it is the solution of heat equation) as


    $$-100x = \sum_{n=1}^{\infty}b_n\sin n\pi x$$


    $$b_n=2 \int_0^1{(-100x)\sin (n \pi x)}dx = -200 \int_0^1 x \sin (n \pi x) dx =(-1)^n \frac{200}{n \pi}$$


    I know it is simple and stupid, but I do not understand how the last integration of this equation yields to $(-1)^n \frac{200}{n \pi}$ (only the last part).


    Can someone explain how this integration is made, and how will be the solution if removing $x$ from the initial equation as


    $$-100 = \sum_{n=1}^{\infty}b_n\sin n\pi x$$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    6,419
    Thanks
    2788

    Re: How to integrate from a Fourier sine coefficient?

    forget about the constant for a moment and let's look at

    $\displaystyle \int_0^1 x \sin(n \pi x)~dx$

    use integration by parts

    $u=x,~du=dx$

    $dv = \sin(n \pi x),~v = -\dfrac{\cos(n \pi x)}{n\pi}$

    $\displaystyle \int_0^1 x \sin(n \pi x)~dx =$

    $\left . -\dfrac{x \cos(n \pi x)}{n\pi}\right |_0^1 + \displaystyle \int_0^1~\dfrac{\cos(n \pi x)}{n\pi}~dx = $

    $-\dfrac{(-1)^n}{n\pi}+\left . \dfrac{\sin(n\pi x)}{n^2\pi^2} \right |_0^1 = $

    $-\dfrac{(-1)^n}{n\pi}$

    now multiply by the constant of -200

    $-200 \displaystyle \int_0^1 x \sin(n \pi x)~dx = 200 \dfrac{(-1)^n}{n\pi}$
    Thanks from HallsofIvy, topsquark and kimia
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2019
    From
    UK
    Posts
    5

    Re: How to integrate from a Fourier sine coefficient?

    Thanks for the detailed reply, but I still do not understand how the integration goes for

    $$-100 = \sum_{n=1}^{\infty}b_n\sin n\pi x$$

    Sorry if it is a stupid question.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Integrating the Fourier coefficient
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Sep 3rd 2017, 05:45 AM
  2. Find Fourier Series Coefficient
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jul 16th 2014, 04:31 AM
  3. Fourier Coefficient
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Jun 2nd 2011, 04:41 PM
  4. PDE-fourier sine series (help!!!)
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 14th 2009, 06:17 AM
  5. Fourier sine series...
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Feb 10th 2008, 08:53 AM

Search Tags


/mathhelpforum @mathhelpforum