# 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.

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sorry i could not enlarge symbols,if you could i will be happy – dato datuashvili Apr 14 '13 at 12:01
thanks @UnkleRhaukus for update – dato datuashvili Apr 14 '13 at 12:17

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.