Section 21.4

Step Functions

Batteries turning on, forces suddenly stopping, and digital signals. We model "sudden changes" with the Unit Step function.

Introduction

Until now, our forcing functions were continuous (exponentials, sines). But real systems have switches.

The Heaviside Function

Definition

The Unit Step Function (or ) is defined as:

It "turns on" at .

Its Laplace Transform:

Visual: The Switch

Interactive: Piecewise Construction

We write a piecewise function on as .

Second Translation Theorem

This relates shifting in to exponentials in .

Form 1: Inverse Friendly

Best for taking Inverse Transform.

Form 2: Forward Friendly

Best for transforming a given function.

Worked Examples

Example 1: Pulse Transform

Transform for and 0 otherwise.

1. Write with Steps:

2. Transform:

3. Combine:

Example 2: Forward Transform (Tricky)

Find .

1. Use Form 2:

2. Evaluate Shifted Function:

3. Transform:

Example 3: Inverse Transform

Find .

1. Identify Parts:

Term means and a shift.

Base function .

2. Apply Shift:

Equation is .

3. Result:

This is a "ramp" function that stays flat at 0 until , then grows.