# Laplace Transform of convolution

• Sep 24th 2010, 11:10 AM
gaurav5
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......
• Sep 24th 2010, 11:15 AM
Ackbeet
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?
• Sep 25th 2010, 06:41 AM
gaurav5
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 .
• Sep 25th 2010, 07:11 AM
gaurav5
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=∞.
• Sep 27th 2010, 04:02 AM
Ackbeet
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!
• Sep 27th 2010, 05:13 AM
gaurav5
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"
• Sep 27th 2010, 06:03 AM
Ackbeet
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!