The following Table of Laplace Transforms is very useful when solving problems in science and engineering that require

Laplace transform. Each expression in the right hand column (the Laplace Transforms) comes from finding the infinite integral that we saw in the Definition of a Laplace Transform section. Time Function f(t) Laplace Transform of f(t)

f(t) =
1

-1

{F(s)}

F(s) = s>0 s>0

{ f(t)}

t (unit-ramp function) tn (n, a positive

s>0 s>a s>0 s>0

eat sin t cos t tng(t), for n = 1, 2, ...

t sin t t cos t
g(at) eatg(t)

s > || s > ||
Scale property

G(s a) Shift property

eattn, for n = 1, 2, ... te-t 1 e-t/T eatsin t eatcos t u ( t)

s>a s > -1 s > -1/T s>a s>a s>0 s>0

u(t a) u(t a)g(t a)

sG(s) g(0) s2 G(s) s g(0) g'(0) sn G(s) sn-1 g(0) sn-2 g'(0) ... g(n1)

(0)

We saw some of the following properties in the Table of Laplace Transforms.

If a is a constant and f(t) is a function of t, then

{a f(t)} = a
Example

{f(t)}

{7 sin t} = 7

{sin t}

[This is not surprising, since the Laplace Transform is an integral and the same property applies for integrals.]

Property 2. Linearity Property

If a and b are constants while f(t) and g(t) are functions of t, then

{a f(t) + b g(t)} = a
Example

{f(t)} + b

{g(t)}

{3t + 6t2 } = 3

{t} + 6

{t2}

Property 3. Change of Scale Property

If

{f(t)} = F(s) then

Example

Property 4. Shifting Property (Shift Theorem)

{eatf(t)} = F(s a)

Example

{e3tf(t)} = F(s 3)
Property 5.

EXAMPLES

Obtain the Laplace transforms of the following functions, using the Table of Laplace Transforms and the properties given above.
(We can, of course, use Scientific Notebook to find each of these. Sometimes it needs some more steps to get it in the same form as the Table).

(a) f(t) = 4t2 Answer (b) v(t) = 5 sin 4t Answer (c) g(t) = t cos 7t Answer DEMONSTRATION of PROPERTY 5: {t f(t)} For example (c), we could have also used Property 5:

with f(t) = cos 7t.

Now So

So

This is the same result that we obtained using the formula.

(d) f(t) = e2 sin 3t Answer DEMONSTRATION OF No 4: SHIFTING PROPERTY For example (d) we could have used:

{eatg(t)} = G(s a)
Let g(t) = sin 3t

So

This is the same result we obtained before for example (d).

(e) f(t) = t4e Answer

-jt

(f) f(t) = te cos 4t Answer (g) f(t) = t2 sin 5t Answer (h) f(t) = t3 cos t = t2(t cos t) Answer

-t

(i) f(t) = cos23t, given that Answer

We saw some of the following properties in the Table of Laplace Transforms. 1. 2. 3. Time Displacement Theorem: If then

If function f(t) is: Periodic with period p > 0, so that f(t + p) = f(t), and

f1(t) is one period (i.e. one cycle) of the function, written using Unit
Step functions, then

NOTE: In English, the formula says: The Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided -sp by (1 e ).

We first saw the following properties in the Table of Laplace Transforms.

1. If G(s) =

{g(t)}, then

2. For the general integral, if

is the value of the integral when t = 0, then:

EXAMPLES

Use the above information and the Table of Laplace Transforms to find the Laplace transforms of the following integrals: (a) Answer

(b) Answer

(c) Answer

(d) Answer

Definition
Later, on this page... Partial Fraction Types Integral and Periodic Types

If G(s) =
-1

{g(t)}, then the inverse transform of G(s) is defined as:

G(s) = g(t)

Some Properties of the Inverse Laplace Transform

We first saw these properties in the Table of Laplace Transforms.

{a G1(s) + b G2(s)} = a g1(t) + b g2(t)

Property 2: Shifting Property

If

-1

G(s) = g(t), then

-1

G(s - a) = eatg(t)

Property 3

If

-1

G(s) = g(t), then

Property 4

If

-1

G(s) = g(t), then

-1

{e-asG(s)} = u(t - a) g(t - a)

EXAMPLES

Find the inverse of the following transforms and sketch the functions so obtained. (a) Answer (b) Answer (c) Answer

(d) Answer (e) Answer (f) Answer

(g) Answer

(where T is a constant)

Examples Involving Partial Fractions

We first met Partial Fractions in the Methods of Integration section. You may wish to revise partial fractions before attacking this section. Obtain the inverse Laplace transforms of the following functions:

(h) Answer

(i) Answer
Integral and Periodic Types

(j) Answer

(k) Answer

(l) Answer