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Dirac delta; fourier representation

  1. Dec 11, 2017 #1
    1. The problem statement, all variables and given/known data
    I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ##

    But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L?

    2. Relevant equations


    3. The attempt at a solution
    I get the feeling the solution, if one exists, will be in the form ##\frac{2\pi}{L}## but I'm not sure if this is right,

    Many thanks
     
  2. jcsd
  3. Dec 11, 2017 #2

    BvU

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    Yes.

    The outcome is still ##2\pi\delta(k-a)##, though. the integral over the remainder of the domain gives zero. Check out distributions

    Edit, sorry, too quick. a can not lie in ##[0,L]##
     
  4. Dec 11, 2017 #3

    BvU

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    So you get a transform of the (sin x)/x kind (#6 here). Something that in the limit goes towards a delta function.
     
  5. Dec 11, 2017 #4
    Why can a not lie in [0,L]?
     
  6. Dec 11, 2017 #5

    George Jones

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    To see what happens, why don't you just do the integral? Hint: if you feel uneasy integrating an imaginary exponential, use ##e^{i\theta}= \cos\theta + i \sin\theta##.
     
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