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Laplace Transform: Order of transformation

  1. Jan 29, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the Laplace transform for [tex]\[t\left( {\int\limits_0^{t - a} {f(u)du} } \right)H(t - a)\][/tex] proving the properties used.


    2. Relevant equations

    If f(t) transforms into F(s):

    a) [tex]\[{( - t)^{n}f(t) \to {F^{(n)}}(s)\][/tex]

    b) [tex]\[\int\limits_0^t {f(u)du} \to \frac{{F(s)}}{s}\][/tex]

    c) [tex]\[f(t - a)H(t - a) \to {e^{ - as}}F(s)\][/tex]

    3. The attempt at a solution

    Here's my question: in what order should I use these properties? There's only one posibility for the translation, and that's that the transform is multiplied by e-as. But with the derivative of F(s) and the division for s there are two posibilities:

    1) That the Laplace transform of [tex]\[t\left( {\int\limits_0^{t - a} {f(u)du} } \right)H(t - a)\][/tex] be [tex]\[\frac{{ - {e^{ - as}}{F^'}(s)}}{s}\][/tex].

    2) That the Laplace transform be [tex]\[ - {e^{ - as}}{\left[ {\frac{{F(s)}}{s}} \right]^'}\][/tex].

    What's the correct one and why? I couldn't find a reason to use either one but not the other.

    EDIT: In property a), the lineal variable t should have an exponent without brackets, since it's not the n-th derivative but the n-th power.
     
  2. jcsd
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