73 Properties of the Laplace transform

(i) The Laplace transform exists for all $s>\beta$, and $|ℒ(f)(s)|\le \frac{M}{s-\beta }$ for all $s>\beta$.

(ii) The Laplace transform is linear so, that if $f,g$ satisfy $(E)$, then for all $\lambda ,\mu \in 𝐂$ the function $\lambda f+\mu g$ also satisfies $(E)$ and

$$ℒ(\lambda f+\mu g)(s)=\lambda ℒ(f)(s)+\mu ℒ(g)(s).$$

(iii) $xf(x)$ also satisfies $(E)$ and $ℒ(f)(s)$ is differentiable with

$$ℒ(xf(x))(s)=-\frac{d}{ds}ℒ(f)(s).$$

(iv) If $f$ is continuously differentiable and $f^{\prime }$ satisfies $(E)$, then $f$ also satisfies $(E)$ and $ℒ(f^{\prime })(s)=sℒ(f)(s)-f(0).$