Compare Fourier and Laplace transform 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.
 A: 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.
A: Laplace transform convergence is much less delicate because of it's exponential decaying kernel exp(-st), Re(s)>0. 
Laplace transform is an analytic function of the complex variable and we can study it with the knowledge of complex variable.
Laplace is also only defined for the positive axis of the reals.
A: you can as well consider $\omega \in \mathbb{C}$ and talk about a "generalized" Fourier tranform, converging in the appropriate domain of $Im(\omega)$, usually a "strip" parallel to the real axis. You may want to check Lukacs 1970, Th. 7.1.1 or Titchmarsh, E.C. (1975): Introduction to the Theory of Fourier Integrals, Oxford University Press. Reprint of the 1948 second edition.
A: Fourier transform is used to solve the problems on the real line while the Laplace transform is used to solve the problems in the complex plane.
