# Transforms of Discontinuous and Periodic Functions

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.6 – Transforms of Discontinuous and Periodic Functions

## Expected Educational Results

• Objective 23–1: I can sketch step and window functions.
• Objective 23–2: I can evaluate the Laplace transform of step and window functions.
• Objective 23–3: I can sketch periodic functions.
• Objective 23–4: I can evaluate the Laplace transform of periodic functions.
• Objective 23–5: I can evaluate the Laplace transform of the Gamma function.

## Transforms of Discontinuous and Periodic Functions

### Rectangular Window Function

#### Definition: Rectangular Window Function

The rectangular window function} $\Pi_{a,b}(t)$ is defined by

$\Pi_{a,b}(t)=u(t-a)-u(t-b)=\begin{cases} \hfill 0, \hfill & tb \\ \end{cases}$

#### Investigation 05

Sketch the following functions. Explain.

1. $f(t)=\Pi_{1,3}(t)$
2. $f(t)=\Pi_{-2,1}(t)$

#### Investigation 06

Sketch the following functions. Explain.

1. $f(t)=t^2\,\left[u(t)-u(t-9)\right]$
2. $f(t)=\cos{(t)}\,\Pi_{\frac{\pi}{2},\frac{5\pi}{2}}(t)$
3. $f(t)=t^2\,\Pi_{2,3}(t)$

#### Investigation 07

Evaluate the following Laplace transforms.

1. $\mathcal{L}\left\{\Pi_{1,3}(t)\right\}(s)$
2. $\mathcal{L}\left\{e^t\,\Pi_{1,2}(t)\right\}(s)$
3. $\mathcal{L}\left\{\sin{(t)}\,\Pi_{0,\frac{\pi}{4}}(t)\right\}(s)$
4. $\mathcal{L}\left\{\Pi_{a,b}(t)\right\}(s)$, where $a

#### Investigation 08

Find the Laplace transforms of the following functions.

1. $f(t)=\begin{cases} \hfill t^2, \hfill & 0\pi \\ \end{cases}$
2. $f(t)=\begin{cases} \hfill t, \hfill & 05 \\ \end{cases}$