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Math Help - laplace inverse

  1. #1
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    laplace inverse

    find laplace inverse of log(s+a)/(s+b)

    do we have to apply power series in it
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  2. #2
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    Re: laplace inverse

    If you just want the answer, you can get that here:

    Inverse Laplace transform

    - Hollywood
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    Re: laplace inverse

    Quote Originally Posted by prasum View Post
    find laplace inverse of log(s+a)/(s+b)

    do we have to apply power series in it
    You can use the fact that

    \mathcal{L}(tf(t))=\int_{0}^{\infty}e^{-st}[tf(t)]dt=\int_{0}^{\infty}-\frac{d}{ds}e^{-st}f(t)dt=-\frac{d}{ds}F(s)

    If you take the inverse transform of the above you get

    -tf(t)=\mathcal{L}^{-1}\left(\frac{d}{ds}F(s)\right) \implies f(t)=-\frac{1}{t}\mathcal{L}^{-1}\left(\frac{d}{ds}F(s)\right)


    Using this gives

    F(s)=\ln\left(\frac{s+a}{s+b}\right) \implies F'(s)=\frac{1}{s+a}-\frac{1}{s+b}

    f(t)=-\frac{1}{t}\left(e^{-at}-e^{-bt} \right)=\frac{e^{-bt}-e^{-at}}{t}
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