I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is following $$F(\omega)=\int\limits_{-\infty}^\infty f(t)e^{-j\omega t}\mathrm dt$$
and Laplace transform is following one
$$F(s)=\int\limits_{-\infty}^\infty f(t)e^{-st}\mathrm dt$$ where $s=\alpha+j\omega$.
Let us this notation, I can't print symbols exactly, but if we put into equation of Laplace, we will get that because of
$e^{-a-j\omega}=e^{-a}*e^{-j\omega}$.
We get that in integral first function $f(t)$ is multiplied by factor $e^{-at}$ if we put notation of $s$ into Laplace integral and also multiply it by $t$ ,which of course would be some another real function for example $M(t)$ and again it would be back to Fourier transform of this $M(t)$ function . So let us make it more detailed.in Fourier transform we have $e^{-j\omega t}$,in Laplace we have $e^{-st}$ where again $s=\alpha+j\omega$.
If we put this into Laplace, we get
$f(t)e^{-\alpha t-j\omega t}$
which we can write as
$(f(t)e^{-\alpha t})e^{-j\omega t}$,
but first one is real right? And again we get real transform of function, or we can assign $(f(t)e^{-\alpha t})=M(t)$.
I need to clarify main difference between these two transform.
