1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derivative of an integral containing a Dirac delta

  1. Oct 1, 2005 #1
    If I had a function g(x) defined by

    [tex]g(x) = \int_{-\infty}^{\infty} f(x) \delta(x) dx[/tex]

    where [tex]\delta(x)[/tex] is the dirac delta function, what would dg(x)/dx be? The fundamental theorem of calculus requires that [tex]f(x) \delta(x)[/tex] needs to be a continuous and differentiable function before I can immediately say that dg(x)/dx = f(x) \delta(x), which is clearly not the case.
  2. jcsd
  3. Oct 1, 2005 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    [tex]g(x) = \int_{-\infty}^{\infty} f(x) \delta(x) dx[/tex]
    is not a function of x! It's derivative is 0. In fact that's true of any definite integral of any integrable function! Perhaps a little more interesting would be:
    What is the derivative of
    [tex]g(x)= \int_{-\infty}^x f(t)\delta(t)dt[/tex]
    but not a whole lot more! If x< 0, g(x)= 0 and the derivative is 0. If x> 0, g(x)= f(0) and the derivative is 0. However, g(x) is not differentiable at 0.

    (In terms of 'distributions' or 'generalized functions', which is what [itex]\delta(x)[/itex] really is, that is differentiable at 0: the derivative of g(x) is [itex]\delta(x)f(x)[/itex].)
  4. Oct 1, 2005 #3

    The Heaviside function,

    [tex] H(x) = \int_{-\infty}^x \delta(t) \: dt [/tex]

    shows what you mean, but how is it that many people define [tex] H(0) = 1/2 [/tex]? I think I'm missing the point...
  5. Oct 2, 2005 #4


    User Avatar
    Homework Helper

    That is because it is nice if
    but if
    the next best thing is if
  6. Oct 2, 2005 #5


    User Avatar
    Staff Emeritus
    Science Advisor

    The value H(0)= 1/2 is convenient but really irrelevant. H(0) is still distcontinuous at 0. Since the crucial point with distributions is their integral properties values at individual values of x are not important.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Derivative of an integral containing a Dirac delta