Inverse Laplace Transform

flouran · 2010-01-16, 00:06

I was looking through some tables of Laplace Transforms on f(t) the other day, and I noticed that in all cases, as $s \to \infty$ , $F(s) \to 0$ . A question that I have been trying to prove is that if $\lim_{s\to\infty}F(s) = 0$ , then does that necessitate whether $F(s)$ can undergo an inverse Laplace transform (i.e. by the Bromwich integral)?

I suspect that the answer is "no", but if anyone has some attempt at a proof I would appreciate it (my idea would be to use Post's inversion formula and utilizing the Grunwald-Letnikov differintegral for evaluating $F^{(k)}\left(\frac{k}{t}\right)$ , but so far this has been futile).

flouran · 2010-01-18, 23:48

Quote:

Originally Posted by flouran

I was looking through some tables of Laplace Transforms on f(t) the other day, and I noticed that in all cases, as $s \to \infty$ , $F(s) \to 0$ . A question that I have been trying to prove is that if $\lim_{s\to\infty}F(s) = 0$ , then does that necessitate whether $F(s)$ can undergo an inverse Laplace transform (i.e. by the Bromwich integral)?

No.

Quote:

Originally Posted by flouran

I suspect that the answer is "no", but if anyone has some attempt at a proof I would appreciate it (my idea would be to use Post's inversion formula and utilizing the Grunwald-Letnikov differintegral for evaluating $F^{(k)}\left(\frac{k}{t}\right)$ , but so far this has been futile).

Well, I managed to finally prove it.

2010-01-16, 00:06	#1
flouran Dec 2008 833₁₀ Posts	Inverse Laplace Transform I was looking through some tables of Laplace Transforms on f(t) the other day, and I noticed that in all cases, as $s \to \infty$ , $F(s) \to 0$ . A question that I have been trying to prove is that if $\lim_{s\to\infty}F(s) = 0$ , then does that necessitate whether $F(s)$ can undergo an inverse Laplace transform (i.e. by the Bromwich integral)? I suspect that the answer is "no", but if anyone has some attempt at a proof I would appreciate it (my idea would be to use Post's inversion formula and utilizing the Grunwald-Letnikov differintegral for evaluating $F^{(k)}\left(\frac{k}{t}\right)$ , but so far this has been futile).

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