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Table 12.1 LAPLACE TRANSFORM PAIRS
Item Number f ( t ) LLLL [ f ( t )] = F(s)
(^1) K δ( t ) K
2 Ku ( t ) or K K s
3 r ( t ) (^1) s^2
(^4) tnu ( t ) n! sn^ +^1
(^5) e–atu ( t ) 1 ( s + a )
(^6) te–atu ( t ) (^) 1 ( s + a )^2
(^7) tne–atu ( t ) n! ( s + a ) n +^1
(^8) sin(ω t ) u ( t ) ω s^2 + ω^2
(^9) cos(ω t ) u ( t ) s s^2 + ω^2
(^10) e – at sin(ω t ) u ( t ) ω ( s + a )^2 + ω^2
(^11) e – at cos(ω t ) u ( t ) ( s^ +^ a ) ( s + a )^2 + ω^2
(^12) t sin(ω t ) u ( t ) 2 ω s ( s^2 + ω^2 )^2
13 t cos(ω t ) u ( t ) (^) s^2 − ω^2 ( s^2 + ω^2 )^2
(^14) sin(ω t + φ) u ( t ) s^ sin(φ)^ +^ ω^ cos(φ) s^2 + ω^2
(^15) cos(ω t + φ) u ( t ) s^ cos(φ)^ −^ ω^ sin(φ) s^2 + ω^2
(^16) e–at [sin(ω t ) – ω t cos(ω t )] u ( t) 2 ω^3 [( s + a )^2 + ω^2 ]^2
ECE20200, Summer 2012 EXAM II (Aung Kyi San)
(^17) te – at sin(ω t ) u ( t ) (^2) ω s^ +^ a [( s + a )^2 + ω^2 ]^2
18 e − at^ C 1 cos(ω t ) +
C 2 − C 1 a ω
sin(ω t )
^
^
u ( t )
C 1 s + C 2
( s + a )^2 + ω^2
Table 12.2 LAPLACE TRANSFORM PROPERTIES
Property Transform Pair
Linearity L [ a 1 f 1 ( t ) + a 2 f 2 ( t )] = a 1 F 1 ( s ) + a 2 F 2 ( s )
Time Shift (^) L [ f ( t – T ) u ( t – T )] = e – sTF ( s ), T > 0
Multiplication by t (^) L [ tf ( t ) u ( t )] = – d ds
F ( s )
Multiplication by tn L[ tn^ f ( t )] = (−1) n^
dnF ( s ) dsn
Frequency Shift (^) L [ e–atf ( t )] = F ( s + a )
Time Differentiation (^) L d dt
f ( t ) ^
^
= sF ( s ) – f (0 – )
Second-Order Differentiation L^
d^2 f ( t ) dt^2
= s^2 F ( s ) − sf (0−^ ) − f (1)(0−^ )
n th-Order Differentiation L^
dn^ f ( t ) dtn
= snF ( s ) − sn −^1 f (0−^ ) − sn −^2 f (1)(0−^ )
−K − f ( n −1)(0−^ )
Time Integration
(i) L f ( q ) dq −∞
t
^
= F ( s ) s
f ( q ) dq −∞
0 −
s
(ii) L f ( q ) dq 0 −
t
^
F ( s ) s
Time/Frequency Scaling (^) L [ f ( at )] = 1 a
F
s a
ECE20200, Summer 2012 EXAM II (Aung Kyi San)
