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

Periodic Function

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 $f$.

Graphs of Periodic Functions

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. $f(t)=\sin{(t)}\,\Pi_{0,\frac{\pi}{4}}(t)$, $T=\frac{\pi}{4}$.

Laplace Transforms of Periodic Functions

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

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Last Modified: Sunday, 8 November 2020 22:02 EDT