Laplace transform
Noun
Meaning
Laplace transform (plural Laplace transforms)
(mathematics) an integral transform of positive real function
f
(
t
)
{\displaystyle f(t)}
to a complex function
F
(
s
)
{\displaystyle F(s)}
; given by:
F
(
s
)
=
∫
0
∞
f
(
t
)
e
−
s
t
d
t
.
{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}
Examples
As a holomorphic function, the Laplace transform has a power series representation.
As a result, LTI systems are stable provided the poles of the Laplace transform of the impulse response function have negative real part.
Because the Laplace transform is a linear operator, * The Laplace transform of a sum is the sum of Laplace transforms of each term. :: * The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.
He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.
If g is the antiderivative of f : : then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide.
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.