# 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

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

### Inverse Laplace Transforms

#### Investigation 09

Inverse Laplace transforms are procedurally difficult to compute; however, you should use your knowledge of Laplace transforms to find $f(t)$ for the following Laplace transforms of $f(t)$:

1. $\displaystyle\mathcal{L}\left\{f(t)\right\}(s)=\frac{2}{s^2+4}$.
2. $\displaystyle\mathcal{L}\left\{f(t)\right\}(s)=\frac{1}{s-5}$.
3. $\displaystyle\mathcal{L}\left\{f(t)\right\}(s)=\frac{3}{(s-3)^2+9}$.