Noun
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.}
As a holomorphic function, the Laplace transform has a power series representation. Source: Internet
As a result, LTI systems are stable provided the poles of the Laplace transform of the impulse response function have negative real part. Source: Internet
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. Source: Internet
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. Source: Internet
If g is the antiderivative of f : : then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide. Source: Internet
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. Source: Internet