Laplace transforms

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable



t


{\displaystyle t}
(often time) to a function of a complex variable



s


{\displaystyle 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






L


{
f
}
(
s
)
=

∫

0


∞


f
(
t
)

e

−
s
t



d
t
.


{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

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