# 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

### Unit Step Function

#### Definition: Unit Step Function

The unit step function (a.k.a. Heaviside step function) $u(t)$ is defined by $u(t)=\begin{cases} \hfill 0, \hfill & t<0 \\ \hfill 1, \hfill & t>0 \\ \end{cases}$

#### Investigation 01

Sketch the following step functions.

1. $f(t)=u(t)$
2. $f(t)=5u(t)$
3. $f(t)=-3u(t)$
4. $f(t)=u(t-3)$
5. $f(t)=2u(t-5)$

#### Investigation 02

Sketch the following functions. Explain.

1. $f(t)=t\,u(t-1)$\$
2. $f(t)=e^t\,u(t)$
3. $f(t)=\sin{(t)}\,u(t-\pi)$

### Laplace Transform of the Unit Step Function

#### Investigation 03

Evaluate the following Laplace transforms.

1. Find $\mathcal{L}\left\{u(t)\right\}(s)$
2. Find $\mathcal{L}\left\{u(t-4)\right\}(s)$

#### Investigation 04

Evaluate the following Laplace transforms.

1. $\mathcal{L}\left\{t\,u(t-1)\right\}(s)$
2. $\mathcal{L}\left\{\sin{(t)}\,u(t-\pi)\right\}(s)$