Thread: a question on Laplace Transform definition

Hi. I have this problem I am quite unsure of.

For which complex values of $\displaystyle s$ is the Laplace Transform of $\displaystyle e^{-2t} cos(6t) $ defined?

What I did was simplified it and had it as

$\displaystyle \int_0^\infty e^{(-2-s)t}cos(6t) \,dt $

And from the L.T definitions, as t approaches infitinty, the function should converge to zero.

So is it right by saying $\displaystyle -2-s < 0$ and hence $\displaystyle s > 2$ ? But then again, if I take the limit of $\displaystyle e^{(-2-s)t}cos(6t)$ it doesn't converge zero, rather it is undefined.


Thanks for your help.

Thanks.

If we apply the definition of LT is...

$\displaystyle f(s)=\mathcal {L}\{e^{-2t}\cdot \cos 6t \} = \int_{0}^{\infty} \cos 6t \cdot e^{-(2+s)t}\cdot dt$ (1)

The integral in (1) converges for $\displaystyle Re(s) >-2$ and that is also the domain of $\displaystyle f(s)$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$