1. Chapter 5 Laplace Transform
5.1 Introduction
The Laplace transform is an example of an integral
transform, namely, a relation of the form
(1)
which transforms a given function f(t) into another function
F(s); K(t,s) is called the kernel of the transform, and F(s)
is known as the transform of f(t). The most well known
Integral transform is the Laplace transform, where a = 0,
b =∞, and K(t,s) = e-st. In that case Eq. (1) takes the form
(2)
The Laplace transform of f(t) is also denoted as L{f(t)} or as

2. The basic idea behind any transform is that the given problem
can be solved more readily in the “transform domain”. In this
chapter, we use t as the independent variable and consider the
interval 0 ≦ t <∞ because in most applications of the Laplace
transform the independent variable is the time t, with 0 ≦ t <∞.
5.2 Calculation of the Transform
5.3Properties of the transform
5.4 Application to the solution of differential equations
5.5 Discontinuous forcing functions; Heaviside step function
5.6 Impulsive forcing function; Diarac impulse function
5.7 Additional properties

3. 5.2 Calculus of the transform
The first question to address is whether the transform F(s) of
a given function f(t) exist – that is, whether the integral
(1)
converge.
Exponential order: We say that f(t) is one of exponential order
as t → ∞ if there exist real constant K, c, and T such that
(2)
For all t≧T. That is, the set of functions of exponential order is
the set of functions that do not grow faster than exponentially,
which included most functions of engineering interest.

4. Example 1
Example 2
Example 3
Piecewise continuous: We say that f(t) is piecewise
continuous on a ≦ t ≦ b if there exist a finite number of
points t1, t2,…, tn such that f(t) is continuous on each open
subinterval a < t <t1, t1 < t < t2,…, tN < t < b, and has a finite
limit as t approaches end point from the interior of that
subinterval.

5. Theorem 5.2.1 Existence of the Laplace Transform
Let f(t) satisfy these conditions: (i) f(t) is
piecewise continuous on 0 ≦ t ≦ A, for
every A > 0, and (ii) f(t) is of exponential
order as t → ∞, so that there exist real
constant K, c, and T such that
for all t ≧ T. Then the Laplace transform of f(t),
namely, F(s) given by Eq. (1) exists for all s > c.
Proof:

6. Example 4 Find the Laplace transform of f(t) = 1
Example 5 Find the Laplace transform of f(t) = eat
Example 6 Find the Laplace transform of f(t) = sinat

7. Example 7 Find the Laplace transform of f(t) =
Example 8 Find the Laplace transform of f(t) =sinhat
For more information on the Laplace transform, please see
Laplace transform table in Appendix C. (p. 1271)
Define Laplace transform operator L and inverse transform
operator L-1 as:

8. Where γ is a sufficiently positive real number.
(14)
and
(15)
Where γ is a sufficiently positive real number.

9. Theorem 5.3 Properties of the transform
Theorem Linearity of the transform
Let u(t) and v(t) are any two functions such that the transforms
L{u(t)} and L{v(t)} both exist, then
(1)
for any constants a, b.
Example 1

10. Theorem 5.3.2 Linearity of the inverse transform
Let U(s) and V(s) such that the inverse transforms
L-1{U(s)} and L-1{V(s)} exist, then
(3)
for any constants a, b.
Example 2

11. Theorem 5.3.3 Transform of the derivative
Let f(t) be continuous and f’(t) be piecewise continuous on
0 ≦ t ≦ t0 for every finite t0, and let f(t) be of exponential
order as t→∞ so that there are constants K, c, T such that
for all t > T. hen L{f’(t)} exists for all s > c, and
(8)
Proof:

12. Theorem 5.3.4 Laplace Convolution Theorem
(15)
or equivalently
(16)
Proof:

13. Example 3
Example 4

14. 5.4 Application to solution of differential equations
Example 1

16. Example 2

17. Example 3

18. 5.5 Discontinuous forcing functions: Heaviside step function
Definition: Heaviside step function
(1)
Since H(t) is a unit step at t = 0, H(t-a) is a unit step shifted
to t = a, shown in Fig. (1b).
Fig. 1 Unit step function

19. The rectangular pulse shown in Fig. 2 is denoted as P(t;a,b), we have
(2)
More generally, any piecewise continuous function
(3)
defined on 0 < t < ∞ can be given by the single expression
(4)

20. The function shown in Fig. 4 is called a ramp function which
Example 1
Example 2
The function shown in Fig. 4 is called a ramp function which
can be expressed as
Fig. 4 The ramp function of Example 2

21. The differential equation governing the
(8)
(9)
Example 3 LC Circuit
The differential equation governing the
charge Q(t) on the capacitor in the circuit is
Let R = 0 and E(T) be the rectangular pulse with the
magnitude of E0 and can be expressed as
(11a)
(11b)

23. 5.6 Impulsive Forcing Functions; Dirac Impulse Function (Optional)
Besides forcing functions that are discontinuous, we are interested in ones that are impulsive – that is , sharply focused in time.
(1)
where f(t) is the force applied to the
mass. If the force is due to a hammer
blow, for instance, initiated at time
t = 0, then we expected f(t) to be
somewhat as sketched in Fig. 1a.
However, we do not know the functional
form of f(t) corresponding to such an
event as a hammer blow. Eq. (1) can
be solved once the f is well defined.

24. In working with impulsive forces one
normally tries to avoid dealing with the
detailed shape of f and tried to limit one’s
concern to a global quantity known as the
impulse Ι of f, the area under its graph.
The idea is that if e really is small, then the
response x(t), while sensitive to I, should
be rather insensitive to the detailed shape
of f.
This idea suggests that we replace the
unknown f by a simple rectangular pulse
having the correct impulse as shown in
Fig. 1b: f(t) = I / ε for 0 ≦ t ≦ ε, and 0 for t ＞ ε. With f thus simplified we can proceed to solve for the response x(t). We suppose that since ε is very small, we might as well take the limit of the solution as ε → 0, to eliminate ε.

25. Let us denote such a rectangular pulse having a unit impulse as D(t;e)
(2)
(3)
Prove that
(4)
for any function g that is continuous at the origin.

26. (5)
Suppose that g is continuous on for some positive b. We can assume that ε < b because we are letting ε → 0. Thus, g is continuous on the integration interval , so the mean value theorem of the integral calculus tells us that there is a point τ1 in [0, ε], such that Thus Eq. (5) gives
(6)
(7)
Where is known as the Dirac delta function, or unit impulse function. We can think of as being zero everywhere except at the origin and infinite at the origin, in such a way as to have unite area.

27. It should be understood that the result is g(a) for any limits A,B.
Example 1.
(8)
Since δ(t) is focused at t ＝ 0, it follows that δ(t－ a) is focused at t ＝ a, and (7) generalizes to
(12)
It should be understood that the result is g(a) for any limits A,B.
(13)

28. Example 2 RC Circuit
Considering the charge Q(t) on the capacitor of the RC circuit
is governed by the differential equation RQ’+(1/C)Q = E(T).
Let E(T) be an impulse voltage, with impulse I acting at t = T,
and let Q(0)=Q0.
In the Laplace transform, the independent variable has been the t , so the delta function has represented actions that are focused in time. But the argument of the delta function need not to be time. For instance, if ω(x) is the load distribution on a beam; δ(x－a) represents a point unit load at x＝a.

29. Consider a metal plate loaded by pressing a metal coin against
Fig. 3 Load distribution in a beam
Consider a metal plate loaded by pressing a metal coin against
it at the origin (Fig. 4a). The stress distribution within the plate
can be determined once the load distribution can be prescribed.
The coin will flatten slightly at the point of contact, the load w(x)
will be distributed over a short interval from x = -e to x = e.
Whether one needs to determine the exact w(x) or if it suffices
to represent it simply as an idealized point force of magnitude
P, w(x) = Pd(x). (near field and far field, Saint Venant’s principle)
Fig. 4 Delta function idealization

30. 5.7 Additional Properties
Theorem s-Shift
If L{f(t)} ＝ F(s) exists for s＞s0, then for any real constant a,
Example Determine
(4)
Example We can invert (2s＋1)/(s2 ＋2s＋4) by partial fractions,

31. If L{f(t)} ＝ F(s) exists for s＞s0, then for any constant a＞0
Theorem t-Shift
If L{f(t)} ＝ F(s) exists for s＞s0, then for any constant a＞0
(6)
for s＞s0
(7)
Theorem Multiplication by 1/s
If L{f(t)} ＝ F(s) exists for s＞s0, then
(8)
for s ＞ s0 or, equivalently,
(9)

32. Example 3. To evaluate L－1{1/[s(s2＋1)]}.
Theorem Differentiation with Respect to s
If L{f(t)} ＝ F(s) exists for s＞s0, then
(16)
For s ＞ s0 or, equivalently,
(17)
Example 4. From the known transform
(18)

33. For s ＞ s0 or, equivalently,
Theorem Integration with Respect to s If there is a real number s0 such that L {f(t)} ＝ F(s) exists for s ＞ s0, and limt→0 f(t)/t exists, then
(21)
For s ＞ s0 or, equivalently,
(22)

34. Theorem 5.7.6 Large s Behavior of F(s)
Let f(t) be piecewise continuous on for each finite t0 and of exponential order as t → ∞. Then
F(s) → 0 as s → ∞,
sF(s) is bounded as s → ∞.

35. Theorem Initial-Value Theorem Let f be continuous and f’ be piecewise continuous on for each finite t0, and let f and f’ be of exponential order as t → ∞. Then
(30)
Example The function f shown in Fig. 1 is seen to be periodic with period T = 4.

36. Theorem 5.7.8 Transform of Periodic Function
If f is periodic with period T on and piecewise continuous on one period, then
(37)
Example 8. If f is the sawtooth wave shown in Fig. 2, then
T = 2,

37. Use the Laplace transform to solve