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Poularikas A. D. “Laplace Transforms” The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC,1999

2
Laplace Transforms
2.1 2.2 2.3 2.4 Deﬁnitions and Laplace Transform Formulae Properties Inverse Laplace Transforms Relationship Between Fourier Integrals of Causal Functions and One-Sided Laplace Transforms 2.5 Table of LaplaceTransforms 2.2 Table of Laplace Operations 2.3 Table of Laplace Transforms References Appendix 1
Examples • Inversion in the Complex Plane • Complex Integration and the Bilateral Laplace Transform

2.1 Deﬁnitions and Laplace Transform Formulae
2.1.1 One-Sided Laplace Transform
F( s) =

∫ f (t ) e
0

− st

dt

s = σ + jω

f (t) = piecewise continuous and of exponential order

2.1.2One-Sided Inverse Laplace Transform
f (t ) = 1 2πj
σ+ j ∞

σ− j ∞

∫ F( s) e

st

ds

where the integration is within the regions of convergence. The region of convergence is half-plane σ < Re{s}.

2.1.3 Two-Sided Laplace Transform
F( s) =

−∞

f (t ) e − st dt

s = σ + jω

f (t) = piecewise continuous and of exponential order

2.1.4Two-Sided Inverse Laplace Transform
f (t ) = 1 2πj
σ+ j ∞

σ− j ∞

∫ F( s) e

st

ds

where the integration is within the regions of convergence which is a vertical strip σ1 < Re{s} < σ2.

2.2 Properties
2.2.1 Properties of the Laplace Transform (one sided)
TABLE 2.1 Laplace Transform Properties
1. 2. Linearity L {K1 f1 (t ) ± K 2 f 2 (t )} = L {K1 f1 (t )} ± L{K 2 f2 (t )} = K1 F1 (s) ± K 2 F2 (s) Time derivative d  L  f (t ) = sF(s) − f (0 + )  dt  Higher time derivative  dn  L  n f (t ) = s n F(s) − s n −1 f (0 + ) − s n − 2 f (1) (0 + ) − L − f ( n −1) (0 + )  dt  where f (i)(0+), i = 1,2,…,n – 1 is the ith derivative of f (·) at t = 0+. 4.  Integral with zero initial condition L    Integral with initial conditions L  

3.

∫t

0

 F(s) f (ξ) dξ  = s 

5. 6. 7. 8. 9.

 F(s) f ( −1) (0 + ) + f (ξ) dξ  = where f ( −1) (0 + ) = lim t→0 + s s −∞ 
t

t

−∞

f (ξ) dξ

Multiplication by exponential L {e ± at f (t )} = F(s m a) Multiplication by t L {t f (t )} = − d F(s) ; ds L {t n f (t )} = ( −1) n dn F(s) ds s

Time shifting L { f (t ± λ ) u(t ± λ )} = e ± sλ F(s)   t  Scaling L  f = aF( as) ;   a L { f (ta)} = 1  s F a  a a>0

10. 11.

 t ∆ Time convolution L  f1 (t − τ) f2 ( τ) dτ  L { f1 (t ) ∗ f2 (t )} = F1 (s) F2 (s)  0  Frequency convolution

L { f1 (t ) f 2 (t )} = 1 2 πj

x + j∞

x − j∞

F1 ( z ) F2 (s − z ) dz = 1 {F1 (s) ∗ F2 (s)} 2 πj

where z = x + jy, and where x must be greater than the abscissa of absolute convergencefor f1(t) over the path of integration. 12. 13. 14. Initial value lim f (t ) = lim sF(s) provided that this limit exists.
t→0 + s→∞

Final value lim f (t ) = lim sF(s) provided that sF(s) is analytic on the jω axis and in the right half of the s plane
t →∞ s→ 0 +

 f (t )  Division by t L  =  t 

∫ F(s ′) ds ′
s T − st

15.

f (t) periodic

∫e L { f (t )} =
0

f (t ) dtf (t ) = f (t + T )

1 − e − sT

2.2.2 Methods of Finding the Laplace Transform
1. 2. 3. 4. Direct method by solving (2.1.1). Expand f (t) in power series if such an expansion exists. Differentiation with respect to a parameter. Use of tables.

2.3 Inverse Laplace Transforms
2.3.1 Properties
1. Linearity L−1 {c1 F1 (s) ± c2 F2 (s)} = c1 f1 (t ) ± c2 f2 (t ) 2.Shifting L−1 {F(s − a)} = e at f (t ) 3. Time shifting L−1 {e − as F(s)} = f (t − a) 4. Scaling property L−1 {F(as)} = 1 f t a a t>a a>0 F ( n ) ( s) = d n F( s) ds n

()

5. Derivatives L−1 {F ( n ) (s)} = ( −1) n t n f (t )

6. Multiplication by s L−1 {sF(s) − f (0 + )} = L {sF(s)} − f (0 + ) L {1} = f (1) (t ) + f (0)δ(t )  F( s)  = 7. Division by s L−1    s 

t

f (t ′) dt...

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