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Evaluating integral of delta function

  1. Feb 7, 2009 #1
    1. The problem statement, all variables and given/known data
    Evaluate the integral:
    gif.latex?\int_{\infty%20}^{\infty%20}\delta%20(t+3)e^{-t}dt.gif

    2. Relevant equations
    To integrate this, should one use a dummy variable to get the delta function only of t, then integrate, then substitute back in after integration?
     

    Attached Files:

  2. jcsd
  3. Feb 7, 2009 #2

    Tom Mattson

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    The delta will be a function of the dummy variable, not t. But yes, that is what you want to do.

    It's a definite integral. Why would there be any need to substitute back after integration?
     
  4. Feb 7, 2009 #3

    Dick

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    Substitute u=t+3 and use the definition of the delta function.
     
  5. Feb 7, 2009 #4
    Ok I just want to make sure I'm thinking correctly and logically. If I let t1 = t + 3, then I need to change all the 't' in the function to t1. I know that the Delta function is defined as


    gif.latex?\int_{-\infty%20}^{\infty}%20\delta(t)dt%20=%201.gif

    So if I can get it into this form with t1:

    gif.latex?\int_{-\infty%20}^{\infty}%20\delta(t_{1})dt_{1}%20=%201.gif

    then the delta function should simplify as 1 correct?

    The function would look like this:

    gif.latex?\int_{-\infty%20}^{\infty}%20\delta(t_{1})(e^{-t_{1}+3})dt_{1}.gif

    then simplify to this:

    gif.latex?\int_{-\infty%20}^{\infty}%20(e^{-t_{1}+3})dt_{1}.gif

    then separate the exponential, pull the exponential not containing t1 out of the integral, integrate the exponential containing t1, and should be left with just this:

    gif.latex?e^3.gif

    Does this look right? Or am I not thinking about this right.
     
    Last edited by a moderator: May 3, 2017
  6. Feb 7, 2009 #5

    Dick

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    The definition of the delta function is not that integral of delta(t)*dt=1. Many functions have that property. It's that the integral of f(t)*delta(t)*dt=f(0), if f is a continuous function.
     
    Last edited: Feb 7, 2009
  7. Feb 7, 2009 #6
    Oh ok I see what you mean. So if I have
    gif.latex?\int_{-\infty%20}^{\infty}%20\delta(t_{1})(e^{-t_{1}+3})dt_{1}.gif
    it should still evaluate to e^3 right?
     
  8. Feb 7, 2009 #7

    Dick

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    Yes. e^3. Sorry. I didn't read the post through to the end. But, once you have it in the form delta(t)*f(t)*dt you are done.
     
  9. Feb 7, 2009 #8
    Oh ok excellent! Thanks for the help, this has helped me 'de-mystify' the delta function :)
     
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