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Delta function properties

  1. Jul 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Prove the following
    '()( − ) = −′()
    -∞δ'(x)*f(x-a) = -f'(a)
    2. Relevant equations
    -∞δ'(x-a)*f(x) = f(a)
    3. The attempt at a solution

    -∞ δ'(x)*f(x-a) = ∫δ(x)*f(x-a)dx-∫f'(x-a)*δ(x)dx = f(-a) - f'(-a)
    Went from 1st to second by integration by parts
    Used integral definition of delta function to go to 3rd part
     

    Attached Files:

  2. jcsd
  3. Jul 14, 2017 #2
    3. The attempt at a solution

    1) ∫-∞ ∞δ'(x)*f(x-a)
    2) = ∫δ(x)*f(x-a)dx-∫f'(x-a)*δ(x)dx
    3) = f(-a) - f'(-a)
    Went from 1st to second by integration by parts
    Used integral definition of delta function to go to 3rd part

    Made above easier to read
     
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