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Hint : use the limit comparison test with . To prove the identity is a simple residue calculation using the keyhole contour.
(Laplace Transform of ) switch the order of integration (reflection formula)
Last edited by Random Variable; Jun 1st 2010 at 07:28 PM.
nevermind
Originally Posted by Bruno J. Hint : use the limit comparison test with . Could you explain in more detail?
Last edited by Random Variable; Jun 1st 2010 at 09:32 PM.
Originally Posted by Random Variable Could you explain in more detail? Let . We know converges absolutely, hence converges if is finite, which is when...
Originally Posted by Bruno J. Let . We know converges absolutely, hence converges if is finite, which is when... Unless I'm totally mistaken, the original integral is convergent for . I'm not sure how that proof shows that.
My mistake! What I wrote only proves convergence when , and indeed the integral is convergent when .
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