The basic properties of the Laplace transform include linearity, time delay, first derivative, second derivative, Nth order derivative, integration, initial value theorem and final value theorem. Two of its most critical applications are to differential equations and convolution. The linearity element of the Laplace transform holds the following equation: a x f(t) + b x g(t)?a x F(s) + b x F(s).
The Laplace transform is employed in physics and engineering to analyze linear time-variant systems, including electrical circuits, optical devices, harmonic oscillations and mechanical systems. Supplied with a simple mathematical or functional definition of the input or output of a system, the Laplace transform offers an optional functional definition that largely simplifies the assessment of the behavior of the system, or produces a new system based on a set of parameters.
Astronomer and mathematician Pierre-Simon Laplace used the z transform, which is similar to the Laplace transform, when working on probability theory. It bears resemblance to the Fourier transform, although it separates a function into its moments. The Fourier transform, on the other hand, describes a function or signal as a series of frequencies.
The Laplace transform is often defined as a change from the time domain, where outputs and inputs are functions of time, to the frequency domain, in which the same parameters and output are functions of an intricate angular frequency.