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Math Help - a question on Laplace Transform definition

  1. #1
    lpd
    lpd is offline
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    a question on Laplace Transform definition

    Hi. I have this problem I am quite unsure of.

    For which complex values of s is the Laplace Transform of e^{-2t} cos(6t) defined?

    What I did was simplified it and had it as

    \int_0^\infty e^{(-2-s)t}cos(6t) \,dt

    And from the L.T definitions, as t approaches infitinty, the function should converge to zero.

    So is it right by saying -2-s < 0 and hence s > 2 ? But then again, if I take the limit of e^{(-2-s)t}cos(6t) it doesn't converge zero, rather it is undefined.


    Thanks for your help.

    Thanks.
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  2. #2
    MHF Contributor chisigma's Avatar
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    If we apply the definition of LT is...

    f(s)=\mathcal {L}\{e^{-2t}\cdot \cos 6t \} = \int_{0}^{\infty} \cos 6t \cdot e^{-(2+s)t}\cdot dt (1)

    The integral in (1) converges for Re(s) >-2 and that is also the domain of f(s)...

    Kind regards

    \chi \sigma
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