TABLE OF COMMON LAPLACE TRANSFORMS
Time Function f (t ) Unit Impulse δ (t ) Delayed Impulse δ (t − T ) Unit Step Unit Ramp Polynomial Exponential Sine Sin Sinh Cosine Cosine Cosh Damped sine

t≥0

Laplace Transform F (s) 1

u (t ) t tn e − at sin ωt

sin(ωt + φ ) sinh ωt

cosωt cos(ωt + φ ) cosh ωt

e − at sin ωt

Damped cosine

e − at cos ωt

1 t n−1e −at n = 1,2,3,... (n − 1)! 1 (e −at - e −bt ) (b − a ) 1 (ae − at -be −bt ) (a − b) 1 [( z − a )e − at -( z − b)e −bt ] (b − a) e − at e −bt e − ct + + (b − a)(c − a ) (c − b)( a − b) (a − c)(b − c) ( z − a )e − at ( z − b)e − bt ( z − c)e − ct + + (b − a)(c − a ) (c − b)( a − b) (a − c)(b − c) 1 −at e sin(ωt ) ω

e −Ts 1 s 1 s2 n! s n +1 1 s+a ω 2 s +ω2 s sin φ + ω cos φ s2 +ω 2 ω 2 s −ω 2 s 2 s +ω2 s cos φ − ω sin φ s2 +ω 2 s 2 s −ω 2 ω ( s + a) 2 + ω 2 s+a ( s + a) 2 + ω 2 1 (s + a) n 1 ( s + a )( s + b) s ( s + a )( s + b) s+z ( s + a )( s + b) 1 ( s + a )( s + b)( s + c) s+z ( s + a )( s + b)( s + c) 1 (s + a) 2 + ω 2

1 b2

  b − at 1 + ω e sin(ωt − φ )   ω  φ = tan −1   a

1 s[( s + a ) 2 + ω 2 ]

b = a2 + ω 2
 a2 +ω 2    ω2    
1/ 2

(

)

1/ 2

sin(ωt + φ )

s+a s +ω 2
2

ω  φ = tan −1   a ω n −ζω nt e sin (aω n t ) a a2 +ζ 2 = 1 1 −ζω nt 1− e sin (aω n t + φ ) a a2 +ζ 2 = 1 ζ = cos φ

2 ωn , ζ