3.4 Inverse Laplace Transform 199

Table 3.1One-Sided Laplace Transforms

Function of Time Function ofs, ROC

1. 
   

   δ(t) 1, wholes-plane

   
   


2. 
   

   u(t)^1 s,Re[s]> 0

   
   


3. 
   

   r(t) s^12 ,Re[s]> 0

   
   


4. 
   

   e−atu(t),a> (^0) s+^1 a,Re[s]>−a

   
   


5. 
   

   cos( 0 t)u(t) s (^2) +s 2
   0
   ,Re[s]> 0

   
   


6. 
   

   sin( 0 t)u(t) s 2 +^0  2
   0
   ,Re[s]> 0

   
   


7. 
   

   e−atcos( 0 t)u(t),a> (^0) (s+as)+ 2 a+ 2
   0
   ,Re[s]>−a

   
   


8. 
   

   e−atsin( 0 t)u(t),a> (^0) (s+a) (^20) + 2
   0
   ,Re[s]>−a

   
   


9. 
   

   2 A e−atcos( 0 t+θ)u(t),a> (^0) s+Aa−∠jθ 0 +s+Aa∠+−jθ 0 ,Re[s]>−a

   
   


10. 
    

    (N−^11 )!tN−^1 u(t) sN^1 Nan integer,Re[s]> 0

    
    


11. 
    

    (N−^11 )!tN−^1 e−atu(t) (s+^1 a)N Nan integer,Re[s]>−a

    
    


12. 
    

    (N^2 −A 1 )!tN−^1 e−atcos( 0 t+θ)u(t) (s+aA−∠jθ 0 )N+(s+Aa∠+j−θ 0 )N,Re[s]>−a

    
    

Table 3.2Basic Properties of One-Sided Laplace Transforms

Causal functions and constants αf(t),βg(t) αF(s),βG(s)

Linearity αf(t)+βg(t) αF(s)+βG(s)

Time shifting f(t−α) e−αsF(s)

Frequency shifting eαtf(t) F(s−α)

Multiplication byt t f(t) −dFds(s)

Derivative dfdt(t) sF(s)−f( 0 −)

Second derivative d

(^2) f(t)
dt^2 s
(^2) F(s)−sf( 0 −)−f( 1 )( 0 )
Integral
∫t
0 −f(t
′)dt F(s)
s
Expansion/contraction f(αt) α6= (^0) |^1 α|F
(s
α
)
Initial value f( 0 +)=lims→∞sF(s)
Final value limt→∞f(t)=lims→ 0 sF(s)
Simple Real Poles
IfX(s)is a proper rational function
X(s)=
N(s)
D(s)

N(s)
∏
k(s−pk)
(3.21)