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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


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 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
up vote 3 down vote accepted

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.

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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.

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thanks for your help – dato datuashvili May 25 '14 at 15:55

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.

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