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UnkleRhaukus
 4 years ago
Show that
\[\mathcal{L} \{{1 \over x}f(x)\} =\int _s^∞F(s)ds\]
UnkleRhaukus
 4 years ago
Show that \[\mathcal{L} \{{1 \over x}f(x)\} =\int _s^∞F(s)ds\]

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JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2The RHS is equal to \[ \int_s^\infty \int_0^\infty f(x) e^{sx} dx \ ds \] Now swap the order of integration and you'll see it's not hard to show that this must be equal to \[ \int_0^\infty \frac{1}{x}f(x) e^{sx} dx \]

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0how do we swap order of integration again

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2As ever, draw a diagram first and see what the region is. Then figure out how the limits change. In this case, it's pretty straight forward. Try it first and tell if you're stuck.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2figure out how the limits change when you change the order of integration that is.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2They may not change at all; you'll need to convince yourself one way or another.

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0dw:1328840642309:dw

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2Look at the region in x,sspace.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2what's a bit confusing here is that the s of limit of integration wrt to s is actually not the same variable. Which is to say, it would have been better if the RHS had been written as \[ \int_s^\infty F(s') \ ds' \]

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2So look at the region in x,s'space. It's very regular.

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0so x,s' space yeah

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0is this the right region s' from s to ∞ x form 0 to ∞dw:1328841005870:dw

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2Yes, hence when you change the order of integration, do the limits change?

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0well now x gos from 0 to ∞ and s' goes from s to ∞. the limits have not changed

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0ok i think i have it

UnkleRhaukus
 4 years ago
Best ResponseYou've already chosen the best response.0I got there thank you James

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2It's a nice result, and what you might hypothesize, given the Laplace transform of xf(x)
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