When will laplace transform become fourier transformable?

[iVenky]

Can you find the fourier transform for all laplace transforms? I know that we have to substitute s=jw to find out the laplace transform. What I mean is can you do that for all Laplace transforms or is there any specific condition that it should satify to convert the laplace transform to fourier transform?

For eg: I saw this question-

The Laplace transform of a continuous-time signal x(t) is

\[

X(s)=\frac{5-s}{s^2-s-2}

\].

If the Fourier transform of the signal exists, then x(t) is ?

I get the answer as

e^2tu(t) - 2e^-tu(t)

but the answer that was given is

-e^2tu(-t) - 2e^-tu(t)


Help me out

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I think I got it. It's because of the ROC thing

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I mean in that first case it won't converge

 

[ninju]

Laplace transform is a form of fourier transform. Take a signal ,multiply it by e^(sigma t) then take fourier transform, then its laplace transform. We use laplace transform as a more general case with the sigma factor as a variable, just to see for what values of sigma, it converges.

In other words, fourier tranfrom is LT with sigma equals 0. Hence, if the jw axis is in the ROC, then it will have a FT.