Inverse Laplace Transform Help

[GreenPrint]

Homework Statement

Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate.

Homework Equations

The Attempt at a Solution

 

[pasmith]

Homework Statement

Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate.

It depends on what F is. If F has no singularities then the inverse transform
<br /> \mathcal{L}^{-1}\left(\frac{F(s)}{s + a}\right) = \frac{1}{2\pi i}\int_{c-i\infty}^{c + i\infty} \frac{F(s)e^{st}}{s+a}\,ds<br />
(where c \in \mathbb{R} is such that there are no singularities of F(s)/(s+a) to the right of the line \mathrm{Re}(s) = c) reduces to the residue of \dfrac{F(s)e^{st}}{s+a} at s = -a, which is F(-a)e^{-at}.

 

[Ray Vickson]

Homework Statement

Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate.

Homework Equations

The Attempt at a Solution

Use convolution: if
f(t) = L^{-1}[F(s)](t),
then
L^{-1} \left( \frac{F(s)}{s+a} \right) (t) = \int_0^t f(t-\tau) e^{-a \tau} \, d \tau.

 

[GreenPrint]

So if you have no idea what one of the functions in the frequency domain is and you get something like

inverse Laplace transform( F(s)G(s) )

and you know what G(s) of is but F(s) is not given then you have no way to evaluate the expression?

 

[Ray Vickson]

So if you have no idea what one of the functions in the frequency domain is and you get something like

inverse Laplace transform( F(s)G(s) )

and you know what G(s) of is but F(s) is not given then you have no way to evaluate the expression?

Of course. If I give you two different F(s), you will get two different answers.