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Dirac Delta Function

  1. Jan 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Evaluate the following integrals:
    [tex]\int^{+\infty}_{-\infty}\delta[f(x)]dx[/tex]
    and
    [tex]\int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx[/tex]

    2. Relevant equations
    [tex]\int^{+\infty}_{-\infty}\delta(x)dx=1[/tex]
    [tex]\int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0)[/tex]
    [tex]\int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)[/tex]


    3. The attempt at a solution
    Part a:
    [tex]\int^{+\infty}_{-\infty}\delta[f(x)]dx[/tex]
    let:
    [tex]u=f(x)[/tex]
    [tex]du=f'(x)dx[/tex]
    [tex]\int^{+\infty}_{-\infty}\delta[f(x)]dx=\int^{+\infty}_{-\infty}\frac{\delta(u)du}{f'(x)}=\int^{+\infty}_{-\infty}\frac{\delta(u)du}{f'[f^{-1}(u)]}=\frac{1}{f'[f^{-1}(0)]}[/tex]
    Is this correct?
     
    Last edited: Jan 15, 2010
  2. jcsd
  3. Jan 15, 2010 #2

    Hurkyl

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    I don't see
    1. How you computed the new limits of integration after substituting
    2. How you determined f was invertible
    3. How you determined f was differentiable
     
  4. Jan 16, 2010 #3
    All three were not specified, I just assumed. Usually when we get questions like these the functions are assumed to be invertible and differentiable. I just copied the question verbatim. Without those assumptions I can't go anywhere... or can I?
     
  5. Jan 17, 2010 #4

    Hurkyl

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    The first one was specified: the original limits of integration were [itex](-\infty, +\infty)[/itex]. When you did the substitution, it looks like you transformed the integrand but forgot to transform the limits.

    What to do if f is not differentiable? I dunno.

    What to do if f is not invertible? This one should be manageable -- in fact, the identity you are tasked to derive is often stated for the non-invertible case! (with an extra assumption on f')


    It's a shame your text doesn't make some statement someplace about what assumptions it's making on functions, either in the exercise or somewhere in an introductory chapter. :frown:
     
  6. Jan 17, 2010 #5
    Well... This is the first homework of the intro E&M class, not math class... so I guess I can't really blame anyone for not making statements.

    Ok, now for the limits:
    If I don't know the function f(x), then how can I set the limits for the integration? Where do you suggest that I look? I looked the the wiki page for the dirac delta but doesn't really help.
     
  7. Jan 17, 2010 #6

    Hurkyl

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    (assuming of differentiable and invertible...)

    You did everything else about the problem without "knowing" the function f(x) -- what is it about the limits that's giving you trouble?
     
  8. Jan 17, 2010 #7
    Well I let u=f(x), so how can I know how f(x) behave at at x=infinity?
     
  9. Jan 17, 2010 #8

    Hurkyl

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    What would you do if you did know f(x)?
     
  10. Jan 17, 2010 #9
    The new limits of integration.
    [tex]\int_{f(-\infty)}^{f(+\infty)}\frac{\delta(u)du}{f'[f^{-1}(u)]}[/tex]
    Now, assuming that u or for that matter f(x) has places which equal zero between the limits of the integral above, the delta function picks out the terms that u=f(x)=0.
    [tex]\int_{f(-\infty)}^{f(+\infty)}\frac{\delta(u)du}{f'[f^{-1}(u)]}=\frac{1}{f'[f^{-1}(u')]}|_{u'=0}[/tex]
     
    Last edited: Jan 17, 2010
  11. Jan 17, 2010 #10

    Hurkyl

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    That looks reasonable. Now, we did make use of the assumption f is invertible -- because you used a substitution. (Also, if f is invertible, its graph cannot cross the x-axis twice)

    But you forgot one thing (albeit an unfortunately common one) -- while Dirac delta does satisfy
    [tex]\int_{-3}^{7} \delta(x) f(x) \, dx = f(0)[/tex]​
    it does not satisfy
    [tex]\int_{4}^{-2} \delta(x) f(x) \, dx = f(0)[/tex]​
    ....




    If f is not invertible, you still seem to have some idea about what the integrand could be. But substitution only works on domains where f is invertible! Can you think of a way to proceed? (You'll probably have to make an additional assumption on f -- so figure out an idea, and then figure out what you assumed to use your idea)
     
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