The Laplace Transformation

The Laplace Transform is a powerful technique for analyzing linear timeinvariantsys tems such as electrical circuits and mechanical systems. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. In actual physical systems the Laplace transform is often interpreted as a transformation from the time-domain (inputs and outputs are functions of time), to the frequency-domain (inputs and outputs are functions of complex angular frequency, or radians per unit time). This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system.

This integral transform has a number of properties that make it useful for analysing linear dynamical systems. A significant advantage is that differentiation and integration become multiplication and division, respectively. (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve. It is a powerful tool for solving a wide variety of initial-value problems.

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Definition if Laplace Transform

In mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, engineering, and probability theory.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform resolves a function or signal into its modes of vibration, the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Denoted \displaystyle\mathcal{L} \left\{f(t)\right\}, it is a linear operator on a function f(t) (original) with a real argument t (t ≥ 0) that transforms it to a function F(s) (image) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s). source : wikipedia.org