79 Holomorphic Laplace transform

Proposition

Suppose that $f$ satisfies (E). Then

(i) ${\cal L}(f)(s)$ defines a holomorphic (complex differentiable) function of $s$ on the open right half plane $\{s\in{\bf C}:\Re s>\beta\}$ ;

(ii) ${\cal L}(f)(s)\rightarrow 0$ as $s\rightarrow\infty$ along $(0,\infty)$ .

Proof. (i) Similar to Proposition 73(iii)

(ii) This is similar to Proposition 73(i).