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In systems theory and signal processing, we often transform expressions based in the Laplace $s$-domain into the complex frequency domain with $j\omega$ (engineering notation for the angular frequency on the imaginary axis): $s \leftrightarrow j\omega$.

I have always been told I can easily transform expressions between the two, but that I shouldn't simply equate them. So far, I always simply considered the Laplace $s$ as a 'sort of frequency or pulsation'. Is this correct?

What exactly is the difference in meaning between these two domains?

PS: Feel free to (re)tag, as I am not familiar with this particular forum's tags.

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up vote 2 down vote accepted

The Laplace transform turns linear differential equations into algebraic ones. Multiplication by $s$ is the operation corresponding to differentiation wrt to $t$ in the other domain. Maybe you should think of it as an operator, not a quantity like a generalized frequency.

In an AC circuit, there's a power source with sinusoidal voltage or current, and other elements that are proportional to it, its derivative, or its integral. Resistors, inductors, capacitors. In this case, $s=jw$ because the source is sinusoidal by design, and the derivative of $e^{jwt}$ is $jw e^{jwt}$. In other applications, we can't assume sinusoids everywhere, and $s$ has nothing to do with frequencies.

You might like Wilbur LePage's book, published by Dover.

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Welcome to the site, Andy. $\LaTeX$ is supported here; you might look at the FAQ. Also, the use of $i$ for $\sqrt{-1}$ might be a little clearer here instead of $j$ (though many are likely to be able to translate). – cardinal Apr 9 '12 at 1:04
Since the question/answer is loooong made, if anyone has anything else to add or detail, I would be deeply interested :) – Rui Pimentel Nov 14 '14 at 18:54

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