Transforms Chapter 2

LAPLACE
TRANSFORMS

CHAPTER 2

, Laplace transforms are not only
easy
to manipulate

SPECIAL FUNCTIONS they also allow us to deal with discontinuous functions
-




{ Unit Step function } I Heaviside function )


0 too
UH )
{
Htt ) a
-

= -




,
t > O




fit ) Htt ) =




B
t
Ht ) t > O
,




I ;
Hit as
aa
=



? is
-




gg#¥t¥÷
soda
inn "
.



: "
activates at a

HHHH al
{ He
-
i
-




,
: tfaa at


me




Consider Htt -


at -



Htt -

b ) where oaaab

activates then
deactivates



at
}
{ I Ibb to
Hit at Htt )




tatM←t
-
-
-
b =




fail )
IIb
at
Htt as Htt b '
-




{ hots to
- - -
-




t

, Examples
fit
{
It ) =
Et<

i<t< 2
1




0
t Z 2




Express ftt ) In terms of the Heaviside function :




fly =
fit ) [ Hlt -



a) -
HH -

b ) ]
=
t [ Hlt -
o ) -



HH -
i )] + 12 .


⇒ [ HCH ) -


Hlt -2 ) ]
=

THH ) -


t Hlt -

i ) + ZHH -

i ) -


ZH ( t 2) . -
THH -
l ) + THH -


2)

THH ) zlt
HIT
i ) I ) I t 2) H It 2)
=
-
- -
+ - -




=
. .



=
¥waut it in this farm ... we 'll see
why
letter