In what sense does the Laplace Transform give components of a signal in something called the 's-domain'?
This article was first posted by Ryder Rude from Math Stack Exchange on Nov. 2, 2019.
Original article
Link: https://math.stackexchange.com/questions/3418622/in-what-sense-does-the-laplace-transform-give-components-of-a-signal-in-somethin/4143863#4143863

Laplace Transform (LT) is a generalization of Fourier transform in the sense it can handle a much wider applications in Engineering, Pure and Applied Math. The big difference it can handle signals which grows in time such as step, ramp or quadratic etc. Another feature of Laplace transform is it can readily solve (Initial Value Problem (IVP) while yield Fourier transform for steady state solution as a special case when the abscissa of convergence is to the left of jw axis.
By superposition principle, the complete response of a linear time invariant system consists of zero state response and zero input response.  So, in a way the LT provides us a component which is known as zero input response. Another way to characterize a LTI system is the following observation.  Let h(t) be the impulse response  of a LTI system.  Then by convolution Theorem, the response of a LTI system to a given input is the convolution of its impulse response and input signal.
Nyquist plot are just tools engineers used daily in their design work. It turns out, as far as system stability is concerned, one only to dig into the Laplace transform X(s) in the s domain instead of the Fourier Transform X(jω). For example, Laplace transform is widely used in Aerospace applications such as aircrafts, missiles and satellites. Thanks to the availability of GPS, satellite/wireless communication and the Internet, their growing importance and influence in our daily life make Cell phones a &quot;must have toy&quot; and account for a lot of popularity among young people as well as grown up.
The basic tool of Laplace transform when mathematicians, physicists and engineers alike realizes they are not just a tool for analysis but also for design. H∞ design, Poles placement Controller Design, Regulator and Observer Design, Stability Analysis are the latest examples of how powerful the tools have become. As adaptive control and nonlinear systems gain popularity, Laplace transforms remains hot, as it offers the first clue on how nonlinear system will behave when we linearize the system around its operating point using Taylor's Series approximation.
Needless to say, Fourier transform retains its major role in areas such as image and speech signal processing and machine vision. As we enter the age of Robotics, Self driving Cars, Artificial intelligence, it will be interesting to see who are the ultimate winners. I suspect both will. So will our mathematicians, physicists, engineers. Doctors, drug researchers and developers, medical researchers. have made significant inroad in advancing gene therapy to control cancer or even cure disease. Hopefully, everybody benefits from the progress and effort we made.