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Laplace transform From Wikipedia, the free encyclopedia Jump to navigationJump to search In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable {\displaystyle t}t (often time) to a function of a complex variable {\displaystyle s}s (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.[1][2][3] For suitable functions f, the Laplace transform is the integral {\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}{\displaystyle {\mathcal {L}}\{f\} (s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.} Contents 1 History 2 Formal definition 2.1 Bilateral Laplace transform 2.2 Inverse Laplace transform 2.3 Probability theory 3 Region of convergence 4 Properties and theorems 4.1 Relation to power series 4.2 Relation to moments 4.3 Computation of the Laplace transform of a function's derivative 4.4 Evaluating integrals over the positive real axis 4.5 Relationship to other transforms 4.5.1 Laplace–Stieltjes transform 4.5.2 Fourier transform 4.5.3 Mellin transform
4.5.4 Z-transform 4.5.5 Borel transform 4.5.6 Fundamental relationships 5 Table of selected Laplace transforms 6 s-domain equivalent circuits and impedances 7 Examples and applications 7.1 Evaluating improper integrals 7.2 Complex impedance of a capacitor 7.3 Partial fraction expansion 7.4 Phase delay 7.5 Statistical mechanics 8 Gallery 9 See also 10 Notes 11 References 11.1 Modern 11.2 Historical 12 Further reading 13 External links History The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory.[4] 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.[5] 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.[6] The theory was further developed in the 19th and early 20th centuries by Mathias Lerch,[7] Oliver Heaviside,[8] and Thomas Bromwich.[9]
The current widespread use of the transform (mainly in engineering) came about during and soon after World War II,[10] replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch,[11] to whom the name Laplace Transform is apparently due. From 1744, Leonhard Euler investigated integrals of the form {\displaystyle z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx}z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx as solutions of differential equations, but did not pursue the matter very far.[12] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

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