# Laplace transform.docx - Laplace transform From Wikipedia...

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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. 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. 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. 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 theory was further developed in the 19th and early 20th centuries by Mathias Lerch, Oliver Heaviside, and Thomas Bromwich.
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 advantages of the Laplace transform had been emphasized by Gustav Doetsch, 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. 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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