2. DEFINITION
The Laplace transform f ( s ) of a function f(t) is defined by:
∞
∫e
− st
f (s) = f (t ) dt
0
TRANSFORMS OF STANDARD FUNCTIONS
f(t) f (s)
1 1
s
e−αt 1
s+α
t
1 −T 1
e
T 1+ sT
1− e −α t α
s (s + α )
te− α t 1
(s + α )2
e −α t − e − β t β −α
( s + α )(s + β )
t 1
s2
tn n!
sn +1
e −α t t n n!
(s + α )n +1
sin ωt ω
s + ω2
2
cosωt s
s + ω2
2

3. f(t) f (s)
e −α t sin ω t ω
(s + α )2 + ω 2
e −α t cos ω t s+α
(s + α )2 + ω 2
1− cosωt ω 2
(
s s2 +ω 2
)
1 1
(sin ω t − ω t cos ω t )
2ω 3
(s 2
+ω 2 2
)
t s
sin ω t
2ω (s 2
+ω 2 2
)
 α  s
e − α t  cos ω t − sin ω t 
 ω  (s + α ) 2
+ω 2
s sin φ + ω cos φ
sin (ω t + φ ) s2 +ω 2
α α 2
+ω 2
e −α t + sin ω t − cos ω t
ω ( s + α )(s 2 + ω 2
)
sin 2 ωt 2ω 2
(
s s 2 + 4ω 2
)
cos2 ωt s 2 + 2ω 2
(
s s 2 + 4ω 2
)
sinh βt β
s −β2
2
cosh βt s
s −β2
2

4. f(t) f (s)
e −α t sinh β t β
( s + α )2 − β 2
e −α t cosh β t s +α
( s + α )2 − β 2
t sinh β t 2β s
(s 2
−β 2 2
)
t cosh β t s2 + β 2
(s 2
−β 2 2
)
1
(β t cosh β t − sinh β t)
1
2β 3
(s 2
−β 2 2
)
Transforms of Special Functions
Unit impulse : δ(t) 1
Unit step : H(t) 1
s
Ramp: tH(t) 1
s2
Delayed Unit Impulse: δ(t-T) e-sT
Delayed Unit Step: H(t-T) e − sT
s
Rectangular Pulse: H(t)-H(t-T) 1− e − sT
s