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Laplace Transforms of Functions

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Following are Laplace transforms for common functions of time. When using the table to take inverse transforms, keep in mind that the time functions are valid only for positive values of t.

Time Domain	Complex-Frequency Domain	Type
δ(t)	1	Impulse
U(t)	1 s	Step
A	A s
t	1 s 2	Ramp
t 2	2! s 3
t n for n = 1, 2, 3,...	n! s n + 1
ε−at	1 s + a	Exponential
tε−at	1 (s + a)2	Damped ramp
t 2ε−at	2! (s + a)3
sin ωt	ω s 2 + ω2	Sine
cos ωt	s s 2 + ω2	Cosine
ε−at sin ωt	ω (s + a)2 + ω2	Damped Sine
ε−at cos ωt	s + a (s + a)2 + ω2	Damped Cosine
ε−at [ B cos ωt +  ( C − aB ω )  sin ωt ]	Bs + C (s + a)2 + ω2
2Kε−at cos (ωt + φ)	Kε jφ s + a − jω  +  Kε −jφ s + a + jω