We begin our study of Laplace transforms with the definition, and we derive the Laplace Transform of some basic functions.

### Introduction to the Laplace Transform

#### Definition of the Laplace Transform

To define the Laplace transform, we first recall the definition of an improper integral.
If is integrable over the interval for every , then the *improper integral of over* is
defined as

*converges*if the limit in (eq:8.1.1) exists; otherwise, we say that the improper integral

*diverges*or

*does not exist*. Here’s the definition of the Laplace transform of a function .

It is important to keep in mind that the variable of integration in (eq:8.1.2) is , while is a parameter independent of . We use as the independent variable for because in applications the Laplace transform is usually applied to functions of time.

The Laplace transform can be viewed as an operator that transforms the function
into the function . Thus, (eq:8.1.2) can be expressed as
The functions and form a *transform pair*, which we’ll sometimes denote
by
It can be shown that if is defined for then it’s defined for all .

#### Computation of Some Simple Laplace Transforms

#### Tables of Laplace transforms

Extensive tables of Laplace transforms have been compiled and are commonly used in applications. The brief table of Laplace transforms in Trench 8.8 will be adequate for our purposes.

We’ll sometimes write Laplace transforms of specific functions without explicitly stating how they are obtained. In such cases you should refer to the table of Laplace transforms.

#### Linearity of the Laplace Transform

The next theorem presents an important property of the Laplace transform.

- Proof
- We give the proof for the case where . If then

#### The First Shifting Theorem

The next theorem enables us to start with known transform pairs and derive others.

#### Existence of Laplace Transforms

Not every function has a Laplace transform. For example, it can be shown that for every real number . Hence, the function does not have a Laplace transform.

Our next objective is to establish conditions that ensure the existence of the Laplace transform of a function. We first review some relevant definitions from calculus.

Recall that a limit exists if and only if the one-sided limits both exist and are equal; in this case, Recall also that is continuous at a point in an open interval if and only if which is equivalent to

For simplicity, we define so (eq:8.1.12) can be expressed as If and have finite but distinct values, we say that has a*jump discontinuity*at , and is called the

*jump*in at the figure below.

If and are finite and equal, but either isn’t defined at or it’s defined but
we say that has a *removable discontinuity* at (see the figure below).This
terminology is appropriate since a function with a removable discontinuity at can be
made continuous at by defining (or redefining)

- (a)
- A function is said to be
*piecewise continuous*on a finite closed interval if and are finite and is continuous on the open interval except possibly at finitely many points, where may have jump discontinuities or removable discontinuities. - (b)
- A function is said to be
*piecewise continuous*on the infinite interval if it’s piecewise continuous on for every .

The figure below shows the graph of a typical piecewise continuous function.

It is shown in calculus that if a function is piecewise continuous on a finite closed interval then it’s integrable on that interval. But if is piecewise continuous on , then so is , and therefore exists for every . However, piecewise continuity alone does not guarantee that the improper integral

converges for in some interval . For example, we noted earlier that (eq:8.1.13) diverges for all if . Stated informally, this occurs because increases too rapidly as . The next definition provides a constraint on the growth of a function that guarantees convergence of its Laplace transform for in some interval .*of exponential order*if there are constants and such that In situations where the specific value of is irrelevant we say simply that is

*of exponential order*.

The next theorem gives useful sufficient conditions for a function to have a Laplace transform.

*not necessary*, for to have a Laplace transform.

### Text Source

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