(A) Basic Laplace transforms

2 2(cos ) sL ts

ωω

=+

2 2(sin )L tsωωω

=+

1

!( )( )

ct nn

nL e ts c +

=−

2 2( sin )( )

ctL e ts c

ωωω

=− + 2 2( cos )

( )ct s cL e t

s cω

ω−

=− +

1 Chew T S MA1506-12 Review of lecture 11

1( ) ,atL e s as a

= >−1

!( )nnnL t

s +=

1(1)Ls

=

(A) Basic Laplace transforms (cont.)

{ } 0

0( ) stL t t eδ −− = { }( ) 1L tδ =

2 Chew T S MA1506-12 Review of lecture 11

( ( ))aseL u t a

s

−− =

( ) 2 2sinhL tsωωω

=−

( ) 2 2cosh sL ts

ωω

=−

1( ( ))L u ts

=

(B) Operational properties

( ( )) '( )L tf t F s= −

( ( )) ( )L f t F s=Let

Then

( )( ( )) ( 1) ( )n n nL t f t F s= −

( )( ) ( )atL e f t F s a= − called s-shifting

called t-Shifting

3 Chew T S MA1506-12 Review of lecture 11

( ( ) ( )) ( )asL f t a u t a e F s−− − =

(C) Transform of derivatives

( ') ( ) (0)L f sL f f= −

2( '') ( ) (0) '(0)L f s L f sf f= − −

4 Chew T S MA1506-12 Review of lecture 11

Let ( )( )

( )x t

v ty t

=

dxdv dt

dydtdt

=

Then

( ) (0)dvL sL v vdt

= −