# s-plane

A real function ${\displaystyle f}$ in time ${\displaystyle t}$ is translated into the s-plane by taking the integral of the function multiplied by ${\displaystyle e^{-st}}$ from ${\displaystyle 0}$ to ${\displaystyle \infty }$ where s is a complex number with the form ${\displaystyle s=\sigma +j\omega }$.
${\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt\;|\;s\;\in \mathbb {C} }$
This transformation from the t-domain into the s-domain is known as a Laplace transform. 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 ${\displaystyle e^{-st}}$ not only catches frequencies, but also the real ${\displaystyle e^{-t}}$ effects as well. Laplace transforms therefore cater not only for frequency response, but decay effects as well. For instance, a damped sine wave can be modeled correctly using Laplace transforms.