Mathematical Fundamentals
133
Such transform is calculated as
Y
L
(s) =
+
-
y(t)e
-st
dt =
+
0
e
s
0
t
e
-st
dt =
+
0
e
-(s-s
0
)t
dt =
-
1
s - s
0
(e
-(s-s
0
)
- e
-(s-s
0
)0
) =
1
s - s
0
,
(49)
and it is convergent for those values of s having real part that is larger than the
real part of s
0
. We have seen in appendix A.7 that the exponential function is an
eigenfunction for the operators derivative and integral, which are fundamental
for the description of physical systems. Therefore, we can easily understand the
practical importance of the transform (49).
###
A central property of the Laplace transform is given by the transformation
of the derivative operator into a multiply by s:
dy(t)
dt
sY
L
(s) - [y(0)] ,
(50)
where the term within square brackets is the initial value in the case that y(t) is
a causal function, i.e. y(t) = 0 for any t < 0. Conversely, the integral is converted
into a division by the complex variable s:
t
-
y(u)du
1
s
Y
L
(s) .
(51)
Since physics describes systems by means of equations containing derivatives
and integrals, these equations can be transformed into polynomial equations by
means of the Laplace transform, and the calculus turns out to be simplified.
Example 2. The second Newton's law states that, for a body having mass
m, the relationship among force f , mass, acceleration a, displacement x, and
time t, can be expressed by
f = ma = m
d
2
x
dt
2
,
(52)
where the notation
d
2
x
dt
2
indicates a second derivative, i.e. the derivative applied
twice. The relation (52) is Laplace-transformed into the polynomial equation
F
L
(s) = s
2
mX
L
(s) - [smx(0) + mx (0)] ,
(53)
where the term within square brackets is determined by the initial condition of
displacement and velocity at time 0.
###
13
In a rigorous treatment, the notation e
-(s-s
0
)
should be replaced by a limiting operation
for t .