Results 1 to 3 of 3

Math Help - Integration term by term and orthogonality

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Integration term by term and orthogonality

    \displaystyle\int_0^L\varphi_m(x)\varphi_n(x) \ dx=0 \ \ \ \text{if} \ m\neq n

    \displaystyle\int_0^L\varphi^2_n(x) \ dx>0

    \displaystyle\int_0^L f(x)\sin\left(\frac{m\pi x}{L}\right) \ dx=\sum_{n=1}^{\infty}b_n\int_0^L\sin\left(\frac{n  \pi x}{L}\right)\sin\left(\frac{m\pi x}{L}\right) \ dx

    Every term is 0 except for when m = n.

    \displaystyle\int_0^L f(x)\sin\left(\frac{m\pi x}{L}\right) \  dx=b_m\int_0^L\left[\sin\left(\frac{m\pi x}{L}\right)\right]^2 \ dx

    \displaystyle f(x)=\sin\left(\frac{3\pi x}{L}\right)

    \displaystyle\int_0^L \sin\left(\frac{3\pi x}{L}\right)\sin\left(\frac{m\pi x}{L}\right) \   dx=b_m\int_0^L\left[\sin\left(\frac{m\pi  x}{L}\right)\right]^2 \ dx

    Looking at the LHS:

    \displaystyle b_m\int_0^L\left[\sin\left(\frac{m\pi  x}{L}\right)\right]^2 \ dx=b_m\frac{1}{2}\int_0^L \left[1-\cos\left(\frac{2m\pi x}{L}\right)\right] \ dx

    \displaystyle b_m\frac{1}{2}\left[x-\frac{L\sin\left(\frac{2m\pi x}{L}\right)}{2m\pi x}\right]_0^L\Rightarrow\frac{L}{2} b_m

    \displaystyle b_n=\frac{2}{L}\int_0^L\sin\left(\frac{3\pi x}{L}\right)\sin\left(\frac{m\pi x}{L}\right) \   dx

    What do I do now? My solution I am obtain is way wrong.
    Last edited by dwsmith; February 23rd 2011 at 03:35 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    In Fourier series the integral is

    <br />
\displaystyle<br />
\int_{-L}^L<br />

    If

    <br />
\displaystyle<br />
\int_{0}^L<br />

    other orthogonality conditions may be.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Is this the solution:

    \displaystyle b_n=\begin{cases}0 & \text{if} \ m\neq 3\\ 1 & \text{if} \ m=3\end{cases}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: February 19th 2011, 11:21 AM
  2. Replies: 0
    Last Post: December 30th 2010, 09:36 AM
  3. Power series, term-by-term integration
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 8th 2010, 02:14 AM
  4. [SOLVED] Ratio of the second term to the first term
    Posted in the Math Topics Forum
    Replies: 4
    Last Post: September 26th 2008, 06:13 AM
  5. Replies: 7
    Last Post: August 31st 2007, 08:18 PM

/mathhelpforum @mathhelpforum