44A10-RulesForLaplaceTransform.tex

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\pmcanonicalname{RulesForLaplaceTransform}
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\pmtitle{rules for Laplace transform}
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\pmtype{Derivation}
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\pmclassification{msc}{44A10}
\pmrelated{TableOfLaplaceTransforms}

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\begin{document}
If\, $\mathcal{L}\{f(t)\} = F(s)$,\, then
\begin{itemize}
\item $\mathcal{L}\{e^{at}f(t)\} \,=\, F(s\!-\!a)$ \quad for\; $s > a$,
\item $\mathcal{L}\{f(\frac{t}{a})\} \;=\; a\,F(as)$ \qquad for\; $a > 0$.\\
\end{itemize}

For deriving these rules, we start from the definition of Laplace transform.\, In the first case, we shall use the notation \,$s\!-\!a = r$:
$$\mathcal{L}\{e^{at}f(t)\} = \int_0^\infty\!e^{-st}e^{at}f(t)\,dt = 
\int_0^\infty\!e^{-(s-a)t}f(t)\,dt = \int_0^\infty\!e^{-rt}f(t)\,dt = F(r) = F(s\!-\!a).$$ 
In the second case, we make the change of variable \,$\frac{t}{a} = u$\, and later use the notation\, $sa = r$:
$$\mathcal{L}\{f(\frac{t}{a})\} = \int_0^\infty\!e^{-st}f(\frac{t}{a})\,dt =
a\!\int_0^\infty\!e^{-sau}f(u)\,du = a\!\int_0^\infty\!e^{-ru}f(u)\,du = aF(r) = a\,F(as).$$
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