# 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

### Gamma Function

#### Definition: Gamma Function, $\Gamma (t)$

The gamma function is defined by $\displaystyle \Gamma (t)=\int_0^{\infty}{e^{-u}u^{t-1}\,du}$, $t>0$.

#### Definition: Laplace Transform of the Gamma Function

$\displaystyle \mathcal{L}\left\{t^r\right\}(s)=\frac{\Gamma (r+1)}{s^{r+1}}$

#### Investigation 12

Find the following Laplace transforms:

1. $\displaystyle \mathcal{L}\left\{t^2\right\}(s)$
2. $\displaystyle \mathcal{L}\left\{e^{3t}t^5\right\}(s)$