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Laplace Transform Properties

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The following table contains the most important properties that are needed in the application of the transform method to dynamic models. The functions of time are valid only for t > 0.

Time Domain	Complex-Frequency Domain
f (t)	F(s)
af (t)	aF(s)
f1(t) + f2(t) − f3(t) + ⋅⋅⋅	F1(s) + F2(s) − F3(s) + ⋅⋅⋅
f (t) + g(t)	F(s) + G(s)
ε−at f (t)	F(s + a)
t f (t)	−  d ds F(s)
f ( t a )	aF(as)
[ f (t − a)]U(t − a)	ε−sa F(s)
f (t)	sF(s) − f (0)
f (t)	s 2F(s) − s f (0) −  f (0)
f (t)	s 3F(s) − s 2f (0) − s  f (0) −  f (0)
d nf dtn   for n = 1, 2, 3,...	s nF(s) − s n − 1f (0) − ⋅⋅⋅ − s f n − 2(0) − f n − 1(0)
∫  t   0  f (λ) dλ	1 s  F(s)
f (0+)	lim s → ∞  sF(s)
f (∞)	lim s → 0  sF(s)