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Homework Help: Calculating Inverse Laplace Using Convolutions

  1. Feb 5, 2008 #1
    1. The problem statement, all variables and given/known data

    We are to use the convolution theorem to compute the inverse Laplace transform of the function

    [tex]L=\frac{1}{s^2 + 16}e^{-2s}[/tex]

    2. Relevant equations
    Using a table, I find that [tex]L^{-1}[\frac{1}{s^2 + 16}] = \frac{1}{4}sin(4t)[/tex]
    [tex]L^{-1}[e^{-2s}] = \delta(t-2)[/tex]

    3. The attempt at a solution
    Using the convolution theorem,

    [tex]L^{-1}[\frac{1}{s^2 + 16}e^{-2s}] = \int_{0}^{t} \frac{1}{4}sin(4\tau)\delta(\tau-2)d\tau[/tex]

    My question is, how do I evaluate that integral? I know that when you have the delta function, e.g. [tex]\delta(t-2)[/tex] times a function f(x), and the limits are negative infinity to infinity, it is just f(2). But what if the limits are 0 to t?
  2. jcsd
  3. Feb 6, 2008 #2
    Can anyone help?
  4. Feb 6, 2008 #3


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    Science Advisor

    [tex]\int f(x)\delta(x-a) dx= f(a)[/tex]
    as long as the interval of integration includes x= a and is 0 if it doesn't. It doesn't have to be from [itex]-\infty[/itex] to [itex]\infty[/itex]

    In particular, with a variable upper limit (and assuming a> x0),
    [tex]\int_{x_0}^t f(x)\delta(x-a) dx[/tex]
    is the function that is 0 if t< a, f(a) if t[itex]\ge[/itex] a.
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