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Questions about the delta function

  1. Mar 8, 2017 #1
    1. The problem statement, all variables and given/known data
    I just have a quick question about the delta function, I'm pretty confident in most other cases but in this simple one I'm not so sure.

    $$\int_{-\infty}^{\infty} \phi (x)\delta (-x)dx$$

    2. Relevant equations


    3. The attempt at a solution

    $$\int_{-\infty}^{\infty} \phi (x)\delta (-x)dx$$
    Using substitution where u=-x and du=-dx:
    $$\int_{-\infty}^{\infty} \phi (-u)\delta (u)(-du) = -\phi (0)$$
    Is this correct?
     
  2. jcsd
  3. Mar 8, 2017 #2

    BvU

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    Wouldn't that yield ##\phi(-0)## instead of ##-\phi(0)## ?
     
  4. Mar 8, 2017 #3
    I'm not entirely sure. I do realize that there should be a negative inside the phi test function but I omitted it because it's zero anyway and I would have had to account for it if the delta function was something like (x-a), but here I didn't. However, when I do the substitution I have to deal with a negative u differential which makes the whole function negative in my mind.
     
  5. Mar 8, 2017 #4

    BvU

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    And the bounds of the integral ?
     
  6. Mar 8, 2017 #5
    Ahh the bounds so when I do the substitution I get:
    $$-\int_{\infty}^{-\infty} \phi (-u)\delta (u)du$$
    Then I change my limits and lose the negative.
    Thank you.
     
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