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Dirac-Delta Function Property

  1. Nov 7, 2013 #1
    I'm not sure how they got the RHS of equation 349:

    where did the |y'(xj)| in the denominator come from?

    According to (343) the RHS is only f(x0) which in this case is the jth term of the sum that gives y(xj) = 0..

    pqayw.png
     
  2. jcsd
  3. Nov 7, 2013 #2

    Simon Bridge

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    ... um... from evaluating the integral?

    I don't think that is what (343) says.

    The integral you are evaluating is centered in a small range about ##y_j=y(x_j)##.
    (343) is for the entire y axis, and is centered about the origin.
     
  4. Nov 10, 2013 #3
    I'm sorry I still don't quite understand how the |y'(xj)| in the denominator came about. How do you evaluate the integral?
     
  5. Nov 10, 2013 #4
    It comes from the Jacobian due to a change of variables done in order to perform the integral
     
  6. Nov 10, 2013 #5
    I still don't quite understand why is there a denominator:

    δ(y(x)) is zero everywhere except at the particular xj where δ(y(x)) = δ(y(xj)) = δ(0) = 1

    Then the integral can be simplified to give:

    ∫ f(x) dx from xj-ε to xj+ε where ε is small enough such that it does not coincide with other solutions of y(xi) = 0 for some xi.

    Then that should give f(xj)∫ dx from xj-ε to xj

    = f(xj) * 1 (∫ dx from xj-ε to xj+ε = 1)
    = f(xj)


    where did the |f'(xj)| in the denominator come from?
     
  7. Nov 10, 2013 #6

    Simon Bridge

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    Isn't it zero everywhere except where ##y=y_j:y_j=y(x_j)##
    (What is y(x)?)

    That means the ##\delta(y)## in the integrand turns into ##\delta(y-y_j)## to stay consistent.
    Maths is a language - what is the math here supposed to be describing?

    Now you can apply the rule.

    Also take note: for a pure math interpretation...
    ... how did they change variables?
     
  8. Nov 11, 2013 #7
    That's all wrong. δ(0) isn't equal to unit. The integral is equal to unit. δ(0) itself is an undefined divergent quantity - infinite.
     
  9. Nov 11, 2013 #8
    They changed from an integral over dx to an integral over dy. There is a Jacobean factor.
     
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