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.
Best Answer
Laplace is generalized Fourier transform. It is used to perform the transform analysis of unstable systems. Simply stating, Laplace has more convergence compared to Fourier.