The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former instead of the latter sense. The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier–Mellin integral, and Mellin’s inverse formula):Įxists (as a proper Lebesgue integral). In these cases, the image of the Laplace transform lives in a space of analytic functions in the region of convergence. The Laplace transform is also defined and injective for suitable spaces of tempered distributions. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L∞(0, ∞), or more generally tempered distributions on (0, ∞). In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range. This means that, on the range of the transform, there is an inverse transform. Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero. From 1744, Leonhard Euler investigated integrals of the formĪn alternate notation for the bilateral Laplace transform is B instead of F. The early history of methods having some similarity to Laplace transform is as follows. The advantages of the Laplace transform had been emphasized by Gustav Doetsch to whom the name Laplace Transform is apparently due. The current widespread use of the transform (mainly in engineering) came about during and soon after World War II replacing the earlier Heaviside operational calculus. The theory was further developed in the 19th and early 20th centuries by Mathias Lerch, Oliver Heaviside, and Thomas Bromwich. Laplace’s use of generating functions was similar to what is now known as the z-transform and he gave little attention to the continuous variable case which was discussed by Niels Henrik Abel. The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions in Essai philosophique sur les probabilités (1814) and the integral form of the Laplace transform evolved naturally as a result History So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. 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. The inverse Laplace transform takes a function of a complex variable s (often frequency) and yields a function of a real variable t (often time). The Laplace transform is invertible on a large class of functions. This perspective has applications in probability theory. This power series expresses a function as a linear superposition of moments of the function. As a holomorphic function, the Laplace transform has a power series representation. Techniques of complex variables can also be used to directly study Laplace transforms. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Laplace transforms are usually restricted to functions of t with t ≥ 0. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. The Laplace transform is similar to the Fourier transform. The transform has many applications in science and engineering. It transforms a function of a real variable t (often time) to a function of a complex variable s (complex frequency). In mathematics, the Laplace transform is an integral transform named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/).
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