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Math Help - Laplace Transform of convolution

  1. #1
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    Laplace Transform of convolution

    I have read the following formula

    limit(x=0 to x=∞)∫e^(-px) dx ( limit (y=x to y=∞) ∫ K(y-x) h(y) dy) =

    (limit(u=0 to u=∞)∫K(u) e^pu du)(limit(y=0 to y=∞)∫h(y) e^(-py) dy)


    i.e laplace transform application

    but the formula doesn't hold true for K(y-x)=e^(-(y-x)) and h(y)=y.

    plz reply if someone can help......
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  2. #2
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    So you're saying that the equation

    \displaystyle{\int_{0}^{\infty}e^{-px}\,dx\int_{0}^{\infty}K(y-x)h(y)\,dy=\int_{0}^{\infty}K(u)e^{pu}\,du\int_{0}  ^{\infty}h(y)e^{-py}\,dy}

    does not hold for K(y-x)=e^{-(y-x)} and h(y)=y.

    Is that correct?
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  3. #3
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    Yes Sir, this is the question I want to ask,the equation does not hold good for K(y-x)=e^(-(y-x)) and h(y)=y .
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  4. #4
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    One modification is there in the equation you wrote:

    The second integral (the one involving K(y-x) and h(y) ) has limit from y=x to y=∞.
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  5. #5
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    All right. So you're saying that you think the equation

    \displaystyle{\int_{0}^{\infty}e^{-px}\,dx\int_{x}^{\infty}K(y-x)h(y)\,dy=\int_{0}^{\infty}K(u)e^{pu}\,du\int_{0}  ^{\infty}h(y)e^{-py}\,dy}

    does not hold for K(y-x)=e^{-(y-x)} and h(y)=y.

    Is this the exact question you're asking?

    I would say that I don't think that equation would hold, in general, at all. On the LHS, you've got p's and x's left over after the integrations. On the RHS, you've only got p's left over after the integrations. A better question might be, does the equation

    \displaystyle{\int_{0}^{\infty}e^{-px}\int_{x}^{\infty}K(y-x)h(y)\,dy\,dx=\int_{0}^{\infty}K(u)e^{pu}\,du\int  _{0}^{\infty}h(y)e^{-py}\,dy}

    hold, in general? Are you sure you didn't mean to ask this? Because this equation has a chance of being correct.

    Let me ask this question: from where did you get these equations that you're trying to disprove? Are they a theorem from somewhere? If so, where?

    Thanks!
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  6. #6
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    Yeah exactly,I doubt at the truth of the second equation you wrote(the one with "dy dx" in the last
    I have read the same equation in the book "INTEGRAL EQUATIONS- A SHORT COURSE" by G.CHAMBERS in the chapter "VOLTERRA INTEGRAL EQUATIONS" under the topic named "CONVOLUTION TYPE KERNELS"
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  7. #7
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    Can you scan in and post the relevant page or two of the book that has the equation you're doubting? The second equation I wrote there definitely does not hold for the candidate K and h. Thanks!
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