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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 Laplace Transforms 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) =
…ver más…

Partial fraction method: Any rational function P(s)/Q(s) where P(s) and Q(s) are polynomials, with the degree of P(s) less than that of Q(s), can be written as the sum of rational functions, known as partial fractions, having the form A/(as + b)r, (As + B)/(as2 + bs + c)r, r = 1,2,… . 2. Expand F(s) in inverse powers of s if such an expansion exists. 3. Differentiation with respect to a parameter. 4. Combination of the above methods. 5. Use of tables. 6. Complex inversion (see Appendix 1).

2.4 Relationship Between Fourier Integrals of Causal Functions and One-Sided Laplace Transforms
2.4.1 F (ω) from F (s)
F (ω ) =

∫e
0

− jωt

f (t ) dt

 f (t ) f (t ) =   0

t≥0 t 0, then F(ω) does not exist; the function f (t) has no Fourier transform. c) Let σ = 0, F(s) is analytic for s > 0, and has one singular point on the jω axis, hence, F(s) = 1 1 and there we obtain or F(s) = L{e jω ot u(t )}. But F{e jω ot u(t )} = πδ(ω − ω o ) + jω − jω o s − jω o the correspondence F( s) = Also F( s) = 1 (s − jω o ) n F (ω ) = πj n−1 ( n−1) (ω − ω o ) + F(s) s= jω δ (n − 1)! 1 s − jω o F(ω ) = F(s) s= jω = πδ(ω − ω o ) + F(s) s= jω

δ(n–1)(·) = the (n – 1)th derivative.
d) F(s) has n simple poles jω1, jω2,…, jωn and no other singularities in the half plane Re s ≥ 0. F(s) takes the form F(s) = G(s) + correspondence is

∑ s − jω an n =1

n

where G(s) is free of singularities for Re s ≥ 0. The

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