Topic for Advance Mathematics prepared by our Professor.

1. The Laplace Transform
Felix V. Garde, Jr.
Felix V. Garde, Jr. The Laplace Transform 1 / 13

2. The Transform Concept
One class of transformations, defined by
T{F(t)} =
∞
−∞
K(s, t)F(t)dt
is called integral transform. The function K(s, t) called the kernel of the
transformation is defined as,
K(s, t) =
0 for t < 0
e−st for t ≥ 0
Felix V. Garde, Jr. The Laplace Transform 2 / 13

3. Laplace Transform
The Laplace transform of F(t) is denoted by L{F(t)} and is defined by
L{F(t)} =
∞
0
e−st
F(t)dt
The integral is a function of the parameter s called the function f(s).
L{F(t)} = f(s)
Linearity of operator L. If F1(t) and F2(t) have Laplace transforms and if
c1 and c2 are any constants, then
L{c1F1(t) + c2F2(t)} = c1L{F1(t)} + c2L{F2(t)}
Felix V. Garde, Jr. The Laplace Transform 3 / 13

4. Transform of elementary functions
Find L{ekt}.
L{ekt
} =
∞
0
e−st
· ekt
dt
=
∞
0
e−(s−k)t
dt
for s > k
=
−e−(s−k)t
s − k
∞
0
= 0 +
1
s − k
L{ekt
} =
1
s − k
, s > k
Felix V. Garde, Jr. The Laplace Transform 4 / 13

5. Transform of elementary functions
Find L{sin kt}.
L{ekt
} =
∞
0
e−st
· sin kt dt
recall: e
ax
sin mx dx =
eax
(a sin mx − m cos mx)
a2 + m2
+ c
=
e−st(−s sin kt−k cos kt))
s2 + k2
∞
0
= 0 −
1(0 − k)
s2 + k2
L{sin kt} =
k
s2 + k2
, s > k
Felix V. Garde, Jr. The Laplace Transform 5 / 13

6. Definition
Sectionally continuous function
The function F(t) is said to be sectionally continuous over the closed interval
a ≤ t ≤ b if that interval can be divided into a finite number of subintervals
c ≤ t ≤ d such that in each subinterval:
• F(t) is continuous in the open interval c < t < d.
• F(t) approaches a limit as t approaches each endpoint from within the
interval;
lim
t→c+
F(t) and lim
t→d−
F(t)
Sectionally continuous function.
Felix V. Garde, Jr. The Laplace Transform 6 / 13

7. Definition
Functions of exponential order
The function F(t) is said to be exponential order as t → ∞ if constants
M and b and a fixed t-value t0 exist such that
|F(t)| < Mebt
for t ≥ t0
Functions of class A
The function F(t) is said to be a function of class A if
• Sectionally continuous over every finite interval in the range t ≥ 0
• Of exponential order as t → ∞
Felix V. Garde, Jr. The Laplace Transform 7 / 13

8. Transforms of Derivative
The transform of F (t),
L{F (t)} =
∞
0
e−st
F (t)dt
is,
L{F (t)} = s L{F(t)} − F(0)
and the transform of F (t),
L{F (t)} =
∞
0
e−st
F (t)dt
is,
L{F (t)} = s2
L{F(t)} − s F(0) − F (0)
Felix V. Garde, Jr. The Laplace Transform 8 / 13

9. Transforms of Derivative
In general for F(n)(t) of class A, where L{F(t)} = f(s), then
L{F(n)
(t)} = sn
f(s) −
n−1
k=0
sn−1−k
F(k)
(0)
Illustration
L{F(3)
(t)} = s3
f(s) − s2
F(0) − sF (0) − F (0)
L{F(4)
(t)} = s4
f(s) − s3
F(0) − s2
F (0) − s F (0) − F (0)
Felix V. Garde, Jr. The Laplace Transform 9 / 13

10. The Gamma Function
The gamma function Γ(x) is defined by
Γ(x) =
∞
0
eβ
βx−1
dβ
Theorem
For x > 0,
Γ(x + 1) = xΓ(x)
Theorem
For positive integral n,
Γ(n + 1) = n!
Felix V. Garde, Jr. The Laplace Transform 10 / 13

11. Periodic Functions
Supposed that the function F(t) is periodic with period ω
F(t + ω) = F(t)
and completely determined throughout one period, 0 ≤ t < ω, then
L{F(t)} =
ω
0
e−sβ
F(β)dβ
1 − e−sω
Felix V. Garde, Jr. The Laplace Transform 11 / 13

12. Derivatives of Transforms
If F(t) is of class A, from
f(s) =
∞
0
e−st
F(t)dt
then
f (s) =
∞
0
(−t)e−st
F(t)dt
or
f (s) = L{−t F(t)}
Theorem
IF F(t) is of class A, it follows from L{F(t)} = f(s) that for any positive
integer n,
dn
dsn
f(s) = L{(−t)n
F(t)}
Felix V. Garde, Jr. The Laplace Transform 12 / 13

13. Examples
• If n is a positive integer, obtain L{tnekt} from the known L{ekt}
• Find L{t2 sin kt}
• Find L{t2 cos kt}
Felix V. Garde, Jr. The Laplace Transform 13 / 13