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Thread: 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

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

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

    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

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

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

    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 $\displaystyle p$'s and $\displaystyle x$'s left over after the integrations. On the RHS, you've only got $\displaystyle p$'s left over after the integrations. A better question might be, does the equation

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

    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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