# 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

### Definition: Periodic Function

A function $f(t)$ is a periodic function of period $T(\ne 0)$ if $f(t+T)=f(t)$ for all $t$ in the domain of $f$.

#### Investigation 09

Let $f(t)=f(t+T)$ be a periodic function with period, $T$. Sketch the following. Explain.

1. $f(t)=t^2\,\left[u(t)-u(t-9)\right]$, $T=9$.
2. $f(t)=e^t\,\Pi_{0,1}(t)$, $T=1$.
3. $f(t)=\sin{(t)}\,\Pi_{0,\frac{\pi}{4}}(t)$, $T=\frac{\pi}{4}$.

#### Theorem

If $f$ has a period of $T(\ne 0)$ and is continuous on $[0,T]$, then $F(s)=\mathcal{L}\left\{f(t)\right\}(s)=\dfrac{\int_0^T{e^{-st}f(t)\,dt}}{1-e^{-sT}}$.

#### Investigation 10

Prove the above Theorem.

#### Investigation 11

Evaluate the following Laplace transforms.

1. $\mathcal{L}\left\{f(t)\right\}(s)$, if $f(t)=e^t$, $0, and $f(t)$ has period $2$.
2. $\mathcal{L}\left\{f(t)\right\}(s)$, if $f(t)=t^2$, $0, and $f(t)$ has period $1$.
3. $\mathcal{L}\left\{f(t)\right\}(s)$, if $f(t)=\begin{cases} \hfill t, \hfill & 0, and $f(t)$ has period $2$.