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Linearity

The Laplace transform is a linear operator:

$\displaystyle \zbox{\alpha x(t) + \beta y(t) \longleftrightarrow \alpha X(s) + \beta Y(s)}
$



Proof: Let

$\displaystyle w(t) = \alpha x(t) + \beta y(t),
$

where $ \alpha$ and $ \beta$ are real or complex constants. Then

\begin{eqnarray*}
W(s) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& {\cal L}_s\{w\} \mathrel{\stackrel{\mathrm{\Delta}}{=}}{\cal L}_s\{\alpha x(t) + \beta y(t)\}\\ [10pt]
&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \int_0^\infty \left[\alpha x(t) + \beta y(t)\right] e^{-st} dt\\ [10pt]
&=& \alpha \int_0^\infty x(t) e^{-st} dt
+ \beta \int_0^\infty y(t) e^{-st} dt\\ [5pt]
&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \alpha X(s) + \beta Y(s).
\end{eqnarray*}

Thus, linearity of the Laplace transform follows immediately from linearity of integration


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``The Laplace Transform'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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