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I tried doing a quick search on this question and was very surprised that this information feels very obscure as if it is almost never discussed. Complex frequencies appear in many mathematical concepts such as Laplace Transforms and sources mention the rectangular form as $ s = σ + jω $, but fail to actually explain what the rectangular components stand for.

I saw one source mention this.

https://www.quora.com/What-does-the-real-part-of-s-in-laplace-transform-represents

the real part(sigma) is called nepper frequency it control amplitude of function and its unit is nepper/second . and imaginary part(omega) is called oscillation (radian) frequency it control oscillation and its unit is radian/second.

I just decided to ask here to know what those actually mean. Also, if some people might respond with the rectangular components being irrelevant or having little significant application I just really want to ask this for the sake of knowing.

EDIT: I am asking what σ and ω in $ s = σ + jω $ stand for and why those quantities represent the real and imaginary components of the complex frequency. The source which I cited said that σ is nepper frequency and ω is oscillation frequency.

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  • $\begingroup$ s = jω , that's essentially it. There are some details of range of integration and convergence conditions for LT vs FT but in engineering practice just make that substitution. Thus, complex s corresponds to real ω (sinusoids), real s corresponds to complex ω (exponential growth/decay). Note that complex s (just like real ω) always comes in +/- pairs, as it is a consequence of a solution to 2nd order (or more) differential eqn. Units are 1/time, with a factor of 2pi. The angle in complex plane corresponds to phase. [Not totally sure if this all is what you're asking] $\endgroup$
    – Pete W
    Mar 21 at 14:09
  • $\begingroup$ also note that "multiplication by j" is a 90 degree rotation in the complex plane. So IMO the "s" plane is just ω turned on its side ... so the question of "what is real s" is equivalent to "what is complex ω" $\endgroup$
    – Pete W
    Mar 21 at 14:17
  • $\begingroup$ finally, the usage is often different. s plane tends to be used for representation of systems, i.e. poles/zeros, while ω tends to be used for signals going into and out of those systems ... but the way i see it, they're both complex quantity representing units of 1/time and a phase shift $\endgroup$
    – Pete W
    Mar 21 at 14:24
  • $\begingroup$ With rectangular component do you mean the same as the real component of $s$? $\endgroup$
    – fibonatic
    Mar 21 at 15:09
  • $\begingroup$ Just to be clear, I am talking about what sigma (σ) and omega (ω) are in s = σ + jω. $\endgroup$
    – AndroidV11
    Mar 21 at 22:04

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