# Laplace transform

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Laplace transform
a map of a function, as a signal, defined esp. for positive real values, as time greater than zero, into another domain where the function is represented as a sum of exponentials. Cf. Fourier transform.
[1940-45; after P. S. LAPLACE]

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In mathematics, an integral transform useful in solving differential equations.

The Laplace transform of a function is found by integrating the product of that function and the exponential function e-pt over the interval from zero to infinity. The Laplace transform's applications include solving linear differential equations with constant coefficients and solving boundary value problems, which arise in calculations relating to physical systems.

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in mathematics, a particular integral transform invented by the French mathematician Pierre-Simon Laplace (Laplace, Pierre-Simon, marquis de) (1749–1827), and systematically developed by the British physicist Oliver Heaviside (Heaviside, Oliver) (1850–1925), to simplify the solution of many differential equations (differential equation) that describe physical processes. Today it is used most frequently by electrical engineers in the solution of various electronic circuit problems.

The Laplace transform f(p), also denoted by L{F(t)} or Lap F(t), is defined by the integral

involving the exponential (exponential function) parameter p in the kernel K = ept. The linear Laplace operator L thus transforms each function F(t) of a certain set of functions into some function f(p). The inverse transform F(t) is written L−1{f(p)} or Lap−1f(p).

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Universalium. 2010.

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