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!

Delta function of a function with multiple zeros

  1. Aug 27, 2008 #1
    Hi everyone,

    I was wondering how to deal with delta functions of functions that have double zeros.

    For instance, how does one compute an integral of the form

    [tex]\int_{-\infty}^{\infty}dx g(x)\delta(x^2)[/tex]

    where g(x) is a well behaved continuous everywhere function?

    In general how does one find

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

    where f(x) has a finite number of multiple zeros along with some simple zeros. I know that

    [tex]\delta(f(x)) = \sum_{i=1}^{N}\frac{\delta(x-x_{i})}{|f'(x_{i})|}[/tex]

    where [itex]x_{i}[/itex]'s (for i = 1 to N) are simple zeros of [itex]f(x)[/itex] and it is known that [itex]f(x)[/itex] has no zeros of multiplicitiy > 1.

    but this is of course not valid here. Using this, however I could write

    [tex]\delta(x^2-a^2) = \frac{1}{2|a|}\left(\delta(x-a) + \delta(x+a)\right)[/tex]

    But the limit of this as [itex]a \rightarrow 0[/itex] tends to infinity.

    Any ideas?

    Thanks in advance.
  2. jcsd
  3. Aug 27, 2008 #2


    User Avatar
    Science Advisor

  4. Aug 28, 2008 #3
    Thank you haushofer.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Delta function of a function with multiple zeros