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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