Chapter 5.1, Problem 35P

A Proof of Corollary 5.1.7

(a) Starting from equation (7), use the fact that $\left|f(t)\right|$ is bounded on $\left[0,M\right]$ [ since $f(t)$ is piecewise continuous] to show that if $s>\max (a,0)$, then

$\begin{array}{l}\left|F(s)\right|\le \underset{0\le t\le M}{\max }\left|f(t)\right|{\displaystyle \underset{0}{\overset{M}{\int }}}{e}^{-st}dt+K{\displaystyle \underset{M}{\overset{\infty }{\int }}{e}^{-(s-a)t}dt}\\ =\underset{0\le t\le M}{\max }\left|f(t)\right|\frac{1-e^{-sM}}{s}+\frac{K}{s-a}{e}^{-(s-a)M}.\end{array}$

(b) Argue that there is a constant $K_1$ such that

$\frac{K{e}^{-(s-a)M}}{s-a}<\frac{K_1}{s}$

for $s$ sufficiently large.

(c) Conclude that $\frac{K{e}^{-(s-a)M}}{s-a}<\frac{K_1}{s}$ for $s$ sufficiently large, $\left|F(s)\right|\le L/s$,

where $L=\underset{0\le t\le M}{\max }\left|f(t)\right|+K_1$.