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Stumped by Laplace Transform

Discussion in 'Misc Electronics' started by Steven O., Oct 20, 2005.

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  1. Steven O.

    Steven O. Guest

    I am studying the Laplace Transform, and I was rather puzzled to find
    that the standard tables in the two textbooks we have -- and even the
    tables I found at the Wolfram Research site -- do not provide an
    inverse transform for the function F(s) = s.

    In fact, the Wolfram site does not even give the inverse transform for
    F(s) = 1, although my text gives that as the Dirac Delta function of
    t. However, judging from one of the homework problems, the text I
    have seems to clearly imply that the inverse of F(s) = s is the
    derivative with respect to time of the Dirac Delta function of t.
    However (i) I have no idea how they figure that out, and (ii) I can't
    even imagine what the derivative of the Dirac Delta function would be.

    Can anyone help me out on either score?

    Thanks in advance for all replies.

    Steve O.




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  2. The inverse transform for 1 is dirac(t),as you said. For F(s) = s,
    Mupad gives dirac(t,1), which is the first derivative of dirac(t), so
    that concurs with what you said. This makes sense, since multiplication
    by s is differentiation in the frequency domain. The derivative of
    dirac(t) is called a unit doublet, consisting of a spike to plus
    infinity at 0-, and a spike to minus infinity at 0+. This site confirms
    that its Laplace transform is just s:

    http://www2.latech.edu/~sajones/BIEN 225 Web Pages/Special Functions.htm
    Special Functions
     
  3. Sorry, I should have said that multiplication by s in the frequency
    domain corresponds to differentiation in the time domain.
     
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