UNIT STEP FUNCTION. Solution: Example : Ex: Write the following function in terms of the unit step function.

UNIT STEP FUNCTION

Solution:

?)}({ atuL

0

)()}({ dtatueatuL st

Example:

)s

e(0

ats

e asst

a

st dt e 1.

s

e as

Ex: Write the following function in terms of the unit step function

2

1t if t

2

1t1 if t

t0 if

tf

cos

2

1

12

)( 2

Ex1: ?)}1(2

1{ 2 tutL

Ex2: ?)}2

1(

2

1{ 2 tutL

Ex3: = ?

• Ex: Show that L{ f(t)u(t-a) }=e-asL{ f(t+a) }

Proof: L(f(t-a)u(t-a)}=e-asF(s) (1)

Let f(t-a)=g(t) then f(t)=g(t+a), put in (1)

L{g(t)u(t-a)}=e-asL{g(t+a)}, change from g to f simply

L{f(t)u(t-a)}=e-asL{f(t+a)},

Ex: ?)}1(2

1{ 2 tutL

Impulse FunctionDefine the function fk (t-a) as

In terms of unit step functions

Dirac delta function or unit impulse function

Mathematical expression for the unit impulse function

Some properties of the unit impulse function

a) b)

c)

• Ex: Solve Solution: Taking the laplace transform of both sides

y(0)=0, y’(0)=0

Convolution• Convolution of f(t) and g(t), h(t) is defined as:

Convolution Theorem

Let H(s), F(s), and G(s) denote the laplace transforms of h(t), f(t), and g(t).

If h is the convolution of f and g, h=f* g then

H(s)=F(s)G(s)

h(t)=L-1{F(s)G(s)}

Example:

Laplace transform of tf(t)

Page 264a

Continued

Page 264b

Continued

Page 264c

Pages 265-267a

Continued

Pages 265-267b

Continued

Pages 265-267c

Continued

Pages 265-267d

Continued

Pages 265-267e

Continued

Pages 265-267f