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6.40 Basic properties of Laplace transforms

Theorem.

Let $f(x)$ and $g(x)$ be functions whose Laplace transforms exist for all $s>\alpha$ , and let $c$ be a constant. Then ${\mathcal{L}}(f+g)={\mathcal{L}}(f)+{\mathcal{L}}(g)$ and ${\mathcal{L}}(cf)=c{\mathcal{L}}(f)$ .

Proof. The first statement follows from observing that

\int_{0}^{R}e^{-sx}(f(x)+g(x))\,dx=\int_{0}^{R}e^{-sx}f(x)\,dx+\int_{0}^{R}e^{% -sx}g(x)\,dx

and taking limits as $R\rightarrow\infty$ . By the same argument, the second statement comes down to the fact that

\int_{0}^{R}e^{-sx}.cf(x)\,dx=c\int_{0}^{R}e^{-sx}f(x)\,dx.