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In mathematics and engineering, the s-plane is the complex plane on which Laplace transforms are graphed. It is a mathematical domain where, instead of viewing processes in the time domain modeled with time-based functions, they are viewed as equations in the frequency domain. It is used as a graphical analysis tool in engineering and physics.

A real function f in time t is translated into the s-plane by taking the integral of the function multiplied by $$e^{{-st}}$$ from $$0}$$ to $$\infty$$ where s is a complex number with the form $$s=\sigma +i\omega }$$.

$$F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt\;|\;s\;\in \mathbb {C} }$$

This transformation from the t-domain into the s-domain is known as a Laplace transform and the function F(s) is called the Laplace transform of f. One way to understand what this equation is doing is to remember how Fourier analysis works. In Fourier analysis, harmonic sine and cosine waves are multiplied into the signal, and the resultant integration provides indication of a signal present at that frequency (i.e. the signal's energy at a point in the frequency domain). The Laplace transform does the same thing, but more generally. The $$e^{{-st}}$$ not only captures the frequency response via its imaginary $$e^{{-i\omega t}}$$ component, but also decay effects via its real $$e^{{-\sigma t}}$$ component. For instance, a damped sine wave can be modeled correctly using Laplace transforms.

A function in the s-plane can be translated back into a function of time using the inverse Laplace transform

$$f(t)={1 \over 2\pi i}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}F(s)e^{st}\,ds}$$

where the real number $$\gamma$$ is chosen so the integration path is within the region of convergence of F(s). However rather than use this complicated integral, most functions of interest are translated using tables of Laplace transform pairs, and the Cauchy residue theorem.

Analysing the complex roots of an s-plane equation and plotting them on an Argand diagram can reveal information about the frequency response and stability of a real time system. This process is called root locus analysis.

Root locus
State space (controls)
z-transform

Illustration of a mapping from the s-plane to the z-plane
Kevin Brown (2015) Laplace Transforms at Math Pages.