In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. 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 synthesizing a new system based on a set of specifications.

The Laplace transform is an important concept from the branch of mathematics called functional analysis.

In actual physical systems, the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as 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.

The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory.

The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory. The transform was discovered originally by Leonhard Euler, the prolific eighteenth-century Swiss mathematician.

Definition:

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s).

The parameter s is in the general complex:

This integral transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, with s. (This is similar to the way that logarithms change an operation of multiplication of numbers to the addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve.

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The Laplace transform is a powerful tool formulated to solve a wide variety of initial-value problems. The strategy is to transform the difficult differential equations into simple algebra problems where solutions can be easily obtained. One then applies the Inverse Laplace transform to retrieve the solutions of the original problems