082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

21.3 Laplace transforms of some common functions 629
21.3 LAPLACETRANSFORMSOFSOMECOMMON
FUNCTIONS
Determining the Laplace transform of a given function, f (t), is essentially an exercise
inintegration.Inordertosaveeffortalook-uptableisoftenused.Table21.1listssome
common functions and their corresponding Laplace transforms.
Table21.1
TheLaplace transformsofsome common functions.
Function, f (t) Laplacetransform,F(s) Function, f (t) Laplacetransform,F(s)
1
t
1
s
1
s 2
t 2 2
s 3
t n n!
s n+1
e at 1
s −a
e −at 1
s +a
e −at cosbt
sinhbt
coshbt
e −at sinhbt
e −at coshbt
tsinbt
t n e −at n!
(s +a) n+1 tcosbt
sinbt
cosbt
e −at sinbt
b
s 2 +b 2
s
s 2 +b 2
u(t)unitstep
u(t −d)
b
(s+a) 2 +b 2 δ(t) 1
δ(t −d)
s +a
(s+a) 2 +b 2
b
s 2 −b 2
s
s 2 −b 2
b
(s+a) 2 −b 2
s +a
(s+a) 2 −b 2
2bs
(s 2 +b 2 ) 2
s 2 −b 2
(s 2 +b 2 ) 2
1
s
e −sd
s
e −sd
Example21.2 UseTable 21.1 todetermine the Laplace transformofeachofthe following functions:
(a) t 3 (b) t 7
(c) sin4t (d) e
( −2t
t
(e) cos (f) sinh3t
2)
(g) cosh5t
(i) e −t sin2t
(h) tsin4t
(j) e 3t cost