We introduce the unit step function and some of its applications.

### The Unit Step Function

In the next section we’ll consider initial value problems where , , and are constants and is piecewise continuous. In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms.

#### Laplace Transforms of Piecewise Continuous Functions

We’ll now develop the method of Example example:8.4.1 into a systematic way to find the Laplace
transform of a piecewise continuous function. It is convenient to introduce the *unit
step function*, defined as

The step function enables us to represent piecewise continuous functions conveniently. For example, consider the function

where we assume that and are defined on , even though they equal only on the indicated intervals. This assumption enables us to rewrite (eq:8.4.5) as To verify this, note that if then and (eq:8.4.6) becomes If then and (eq:8.4.6) becomesWe need the next theorem to show how (eq:8.4.6) can be used to find .

- Proof
- By definition, From this and the definition of , The first integral on the right equals zero. Introducing the new variable of integration in the second integral yields Changing the name of the variable of integration in the last integral from to yields

Formula (eq:8.4.6) can be extended to more general piecewise continuous functions. For example, we can write as if , , and are all defined on .

The trigonometric identities

#### The Second Shifting Theorem

Replacing by in Theorem thmtype:8.4.1 yields the next theorem.

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)