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This question already has an answer here:

What are the advantages of Laplace Transform vs Fourier Transform in signal theory?

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marked as duplicate by Matt L., Jim Clay, MBaz, Dilip Sarwate, lennon310 Dec 21 '17 at 3:40

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    $\begingroup$ Signal Processing would be a better place for this question. $\endgroup$ – sammy gerbil Dec 19 '17 at 16:58
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    $\begingroup$ I'm voting to close this question as off-topic because it is about signal processing; not about physics. $\endgroup$ – JMac Dec 19 '17 at 17:08
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    $\begingroup$ Thanks Sammy for writing « a better place ». Writing that signal processing is off-topic in Physics is a bit arrogant and uninformed : ask Gabor, Brillouin, Meyer, etc... !!! $\endgroup$ – GeorgK Dec 19 '17 at 17:50
  • $\begingroup$ Laplace and Fourier transforms are an integral part of periodic systems analysis- a cornerstone of physics. $\endgroup$ – LegitimateWorkUser Dec 19 '17 at 17:59
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Laplace transforms can capture the transient behaviors of systems. Fourier transforms only capture the steady state behavior. Of course, Laplace transforms also require you to think in complex frequency spaces, which can be a bit awkward, and operate using algebraic formula rather than simply numbers.

If you want to see the power distribution of a signal over time, Fourier transformations are often the easiest way to do it. However, if you want to understand what a system does when you flip a light switch, you typically need Laplace transforms.

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  • $\begingroup$ The second sentence of this answer is a common misconception. It's not true that the Fourier transform can only capture the steady state behavior. The frequency response of an LTI system, which is the Fourier transform of its impulse response, completely characterizes its behavior, and hence it covers transients as well as the steady state response. $\endgroup$ – Matt L. Dec 20 '17 at 10:46
  • $\begingroup$ Cort (i remember you from the philosophy SE, fondly), i gotta tell you that @MattL. is right. you have a score of 5 on this, but with no help from me. $\endgroup$ – robert bristow-johnson Dec 20 '17 at 17:16
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In addition, the Fourier Transform exists along the $j \omega$ axis of the two sided Laplace Transform so Fourier is a slice of Laplace.

The $\sigma$ axis captures the dissipation/regeneration properties of Linear Systems. While there are Bode criteria for stability in the frequency transfer function, the $\sigma$ axis is very useful in feedback stability analysis like a root locus plot.

The Fourier is used primarily to describe signals while the Laplace is associated with the system that the signal propagates through.

These concepts generalize naturally to higher dimensions such as complex wave number. A $\sigma$ can Model things like skin effects in wave guides.

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Broadly speaking (and over-simplifying), I would say that Laplace Transform (LT) is better-suited for symbolic computation and systems analysis whereas Fourier Transform (FT) is very useful for numerical computation and signal analysis.

One notable differences is the mathematical space on which they operate: FT is well-defined on $L_2$ space where we have Plancherel Theorem (and even on $L_1$ space), whereas the LT is (generally) defined only for functions of $t>0$ (or sometimes, both of them called monolateral LT).

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  • $\begingroup$ I'm afraid all three of your points are inaccurate. The LT is not more general than the FT, it's different, and there are functions that have a FT but no LT (e.g., sinusoids and impulse responses of ideal frequency selective filters). There are numerical algorithms for computing the LT (and its inverse), and the FT can capture any system response, also transients, just like the LT. $\endgroup$ – Matt L. Dec 20 '17 at 12:13
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    $\begingroup$ I didn't know about numerical algorithms for LT (I'll edit my answer), could you point me to some reference, I would be very interested to know more. For the third point, I agree that the "FT can capture any system response" (and have just read your comment of Cort Amon's answer, with which I completly agree); what I was trying to express is that use of LT permits description of systems via their poles/ and zeros, something that I don't think can be obtained via FT (in a analytical sens I mean). Concerning LT being more general as FT, I wasn't trying to be accurate, I will edit that. $\endgroup$ – Klaz Dec 20 '17 at 13:14
  • $\begingroup$ For numerical methods have a look at this and this paper. The forward LT can be (approximately) computed on a grid using a sequence of FFTs, just like the FT can be approximated by one FFT. $\endgroup$ – Matt L. Dec 20 '17 at 13:36
  • $\begingroup$ One more thing: the step function does have a FT: $$\mathcal{F}\{u(t)\}=\pi\delta(\omega)+\frac{1}{j\omega}$$ $\endgroup$ – Matt L. Dec 20 '17 at 13:38
  • $\begingroup$ And I always thought that the step function, not being in $L_2$, had no FT.... I stand corrected (and confused) $\endgroup$ – Klaz Dec 20 '17 at 13:44