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Dirac Delta (Electrodynamics)

  1. Oct 18, 2009 #1

    E&M

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    1. The problem statement, all variables and given/known data

    I am really confused in my electrodynamics class. I have the following function.

    f(x) = [tex]\delta[/tex] (x + [tex]\alpha[/tex] ) + [tex]\delta[/tex](x -[tex]\alpha[/tex])

    How do i convert this into Fourier Tranform ?
    Those are dirac delta functions on either sides of the origin.


    2. Relevant equations

    no clue

    3. The attempt at a solution

    I multiplied f(x) by 1/sqrt(2*pi)*e-ikt and integrated the whole thing.
     
  2. jcsd
  3. Oct 18, 2009 #2
    And what did you get?
     
  4. Oct 18, 2009 #3

    E&M

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    1/[tex]\sqrt{2\pi}[/tex][tex]\int[/tex][tex]\delta[/tex](x+[tex]\alpha[/tex])e-ikx + [tex]\delta[/tex](x-[tex]\alpha[/tex])e-ikx dx

    I tried to integrate those terms by parts (uv substitution).

    [tex]\int[/tex][tex]\delta[/tex](x+[tex]\alpha[/tex]) = 1. Isn't that right ?
    Also I think f(x)[tex]\int[/tex][tex]\delta[/tex](x+[tex]\alpha[/tex]) = f([tex]\alpha[/tex]). Am i right ?
     
  5. Oct 18, 2009 #4
    All you need is the following property:

    [tex]\int\limits_{-\infty}^{+\infty} f(x)\delta\left(x-a\right)=f(a)[/tex]
     
  6. Oct 18, 2009 #5

    E&M

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    so

    [tex]\int[/tex]e-ikx [tex]\delta[/tex] (x - a) = e-ika ?
     
  7. Oct 18, 2009 #6
    Exactly.
     
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