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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.

    "Spying On The College Of Your Choice" -- How to pick the college that is the Best Match for a high school student's needs.
  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: 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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