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Math Help - laplace transform

  1. #1
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    laplace transform

    Use the formula for the derivative of a Laplace transform to show that if:


    F(s)= arctan(a/s)

    then:


    f(t)= (sin(at))/t
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  2. #2
    Senior Member yeKciM's Avatar
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    use inverse Laplace transformation

     \displaystyle \frac {1}{2\pi i} \lim _{n\to \infty} \int _{\gama -iT} ^{\gama +iT} e^{st}F(S) \; ds
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  3. #3
    MHF Contributor chisigma's Avatar
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    One of the fundamental properties os the Laplace Trasform is that if \mathcal{L} \{f(t)\} = F(s) then...

    \displaystyle \mathcal{L} \{\frac{f(t)}{t}\} = \int_{s}^{\infty} F(u)\ du (1)

    ... so that if f(t)= \sin at then \displaystyle \mathcal{L} \{f(t)\} = \frac{a}{s^{2} + a^{2}} so that is ...

    \displaystyle \mathcal {L} \{\frac{\sin a t}{t}\} = \int_{s}^{\infty} \frac{a\ du}{u^{2} + a^{2}} = \frac{\pi}{2} - \tan^{-1} \frac{s}{a} = \cot^{-1} \frac{s}{a} (2)

    Kind regards

    \chi \sigma
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