74 Proof of properties

(i) Let $|f(x)|\leq Me^{\beta x}$ . Then for $0<W<R$

\Bigl{|}\int_{W}^{R}e^{-sx}f(x)\,dx\Bigr{|}\leq M\int_{W}^{R}e^{\beta x}e^{-sx% }\,dx

=\Bigl{[}{{M}\over{\beta-s}}e^{(\beta-s)x}\Bigr{]}_{W}^{R}

={{M}\over{\beta-s}}e^{(\beta-s)R}-{{M}\over{\beta-s}}e^{(\beta-s)W}\rightarrow 0

as $W\rightarrow\infty$ . Also, we can let $R\rightarrow\infty$ and $W\rightarrow 0+$ to get

\Bigl{|}\int_{0}^{\infty}e^{-sx}f(x)\,x\Bigr{|}\leq{{M}\over{s-\beta}}.

(ii) Suppose that $|f(x)|\leq Pe^{ax}$ and $|g(x)|\leq Re^{bx}$ for all $x>0$ . Then with $\beta=\max\{a,b\}$ and $M=|\lambda|P+|\mu|R$ , we have