Chapter 5.5, Problem 26P

Consider the function $p$ defined by

$p\left(t\right)=\{\begin{array}{l}t,0\le t<1,\\ 2-t,1\le t<2,\end{array}$ and $p\left(t\right)$ has period $2$.

(a) Sketch the graph of $y=p\left(t\right)$.

(b) Find $L\left\{p\left(t\right)\right\}$ by noting that $p$ is the periodic extension of the function $h$ in problem $25(\text{c})$ and then using the result of Theorem $\mathrm{5.5.3}$.

(c) Find $L\left\{p\left(t\right)\right\}$ by noting that $p\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}f\left(\xi \right)}d\xi$, where $f$ is the function in Problem $22$, and then using the Theorem $\mathrm{5.2.2}$.

(c) Let $h\left(t\right)=g\left(t\right)-u_1\left(t\right)g\left(t-1\right)$, where $g$ is defined in part (b). Sketch the graph of $y=h\left(t\right)$ and $L\left\{h\left(t\right)\right\}$.

THEOREM $\mathrm{5.5.3}$

If $f$ is periodic function with period $T$ and is piecewise continuous on $\left[0,T\right]$, then $L\left\{f\left(t\right)\right\}=\frac{F_T\left(s\right)}{1-e^{-sT}}=\frac{{\displaystyle \underset{0}{\overset{T}{\int }}{e}^{-st}f\left(t\right)dt}}{1-e^{-sT}}$.

22. $f\left(t\right)=\{\begin{array}{l}1,0\le t<1\\ -1,1\le t<2\end{array}$ and $f\left(t\right)$ has period $2$.

THEOREM $\mathrm{5.2.2}$

Suppose that $f$ is continuous and $f^{\prime }$ is piecewise continuous on any interval $0\le t\le A$. Suppose further that $f$ and $f^{\prime }$ are of exponential order, with $a$ as specified in Theorem $\mathrm{5.1.6}$. Then $L\left\{f^{\prime }\left(t\right)\right\}$ exists for $s>a$, and moreover $L\left\{f^{\prime }\left(t\right)\right\}=sL\left\{f\left(t\right)\right\}-f\left(0\right)$.