How to get the Laplace transform of $t \cdot f(t) \cdot e^t$

Question

Is there a formula to get the Laplace transform of $t \cdot f(t) \cdot e^t$ ? I tried integration, but that got me nowhere, because I'm probably missing something.

Any ideas?

Answer 1

Hint: $$f(t)e^t\stackrel{\mathcal{L}}{\longleftrightarrow}\mathcal{F}(s-1)$$ and $$tf(t)\stackrel{\mathcal{L}}{\longleftrightarrow}-\frac{d}{ds}\mathcal{F}(s)$$

Answer 2

Let $Imz>a$. \begin{eqnarray*} \int_{0}^{\infty }dt\exp [izt]te^{at} &=&\partial _{a}\int_{0}^{\infty }dt\exp [izt]e^{at} \\ &=&\partial _{a}\int_{0}^{\infty }dt\exp [(iz+a)t]=\partial _{a}\frac{-1}{ iz+a}=\frac{1}{(iz+a)^{2}} \end{eqnarray*} Now set $a=1$. For the real Laplace transform replace $Imz$ by $-s$. We see that the Laplace transform is not defined for all $s$.