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The Laplace transform of the product of two functions may be expressed as the complex convolution integral, e (/(')£('))= £iy(")G(s-»)do> (F-12)

Multiplying the Laplace transform of a function by 5 is analogous to differentiating the original function; thus.

*The final value theorem is valid provided that sF(s) is analytic on the imaginary axis and in the right half of the ¿-plane; i.e., it applies only to stable systems.

Dividing the Laplace transform of a function by s is analogous to integrating the original function; thus, e-'(F(S)/S)=f/(u)du (F-14)

Hie inverse Laplace transform of a product may be expressed as the convolution integral t~\F(s)G(s))= f f(t)g(t-r)dr= /' g(t)f(t ~r)dr (F-15)

which may be inverted to yield

A short list of Laplace transforms is given in Table F-l; detailed tables are given by Abramowitz and Stegun [1968], Korn and Korn [1968], Churchill [1958], and Erdelyi, et al., [1954].

Table F-1. Laplace Transforms

flit)

eis)

Bit)

G (S)

df

S F IS) - 110*1

u (t-al+

exp l-a)/i

dt

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