1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Fourier transform

  1. Jun 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Screenshot2012-06-12at23022AM.png



    3. The attempt at a solution

    I don't understand this step. It's got to be some sort of identity that I missed. I also don't understand why the limits of integration change.
     
  2. jcsd
  3. Jun 12, 2012 #2
    Here's an easy way of seeing it:

    Remember that the integral over an even interval of an odd function is zero
    [tex]\int_{-L}^L f(x) dx = 0 [/tex]
    if [itex] f(-x) = -f(x) [/itex].

    You can see fairly easily that [itex] \frac{\sin(\alpha)}{\alpha} [/itex] is an even function and [itex] \sin(\alpha x) [/itex] is an odd function; therefore [itex] \frac{\sin(\alpha) \sin(\alpha x)}{\alpha} [/itex] is odd and it's integral vanishes over an even support interval.
     
  4. Jun 12, 2012 #3
    ok, I understand what you mean, although it took me about 30 minutes to get it. I still understand why the limits of integration change. I also don't understand why 1/pi changes to 2/pi though I think it has something to with the change in the limits of integration.
     
  5. Jun 12, 2012 #4
    For an even function f(-x) = f(x), you can show that [itex] \int_{-L}^L f(x) dx = 2 \int_0^L f(x) dx [/itex]
     
  6. Jun 12, 2012 #5
    cool
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook