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Laplace Transform
In mathematics, theLaplace transformis an integral transform named after its inventor Pierre-Simon Laplace(/ləˈplɑːs/). It transforms a function of a real variablet(often time) to a function of a complex variable s(complex frequency). The transform has many applications in science and engineering. The Laplace transform is similar to the Fourier transform. While the Fourier transform of a function is a complex function of arealvariable (frequency), the Laplace transform of a function is a complex function of acomplex variable. Laplace transforms are usually restricted to functions oftwitht≥ 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variables. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory. The Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variables(often frequency) and yields a function of a real variablet(often 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. So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication.
https://en.wikipedia.org/wiki/Laplace_transform
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