# Laplace Transforms

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

• Section 7.2 – Definition of the Laplace Transform

## Expected Educational Results

• Objective 19–1: I understand the Laplace transform as a map of a function onto a specific class of functions.
• Objective 19–2: I can use the definition of Laplace transform to find the Laplace Transform of a given any function.

## Laplace Transforms

Definition: Laplace Transforms

Let $f(t)$ be a function on $[0,\infty)$. The Laplace Transform of $f(t)$ is the function $F(s)$ defined as

$\displaystyle F(s)=\mathcal{L}\left\{f(t)\right\}(s)\equiv\int_0^{\infty}{e^{-st}f(t)\,dt}$

### Laplace Transforms of Miscellaneous Functions

#### Investigation 05

Find the Laplace transform following functions:

1. $\displaystyle f(t)=t^ne^{at}$, for $n\geq 1$.
2. Explain what happens to the Laplace transform if $s\leq a$.
3. $\displaystyle f(t)=t^2 + e^{-t}$
4. $\displaystyle f(t)=e^{at}\sin{(bt)}$.
5. Explain what happens to the Laplace transform if $s\leq a$.
6. $\displaystyle f(t)=e^{at}\cos{(bt)}$.
7. Explain what happens to the Laplace transform if $s\leq a$.