Laplace Transforms Essentials

CHAPTER 1

THE LAPLACE TRANSFORM

1.1 INTEGRAL TRANSFORMS

A class of transformations, which are called Integral transforms, are defined by

(1)

Given a function K(s, t), called the kernel of the transformation, equation (1) associates with each function f(t), of the class of functions for which the above exists, a function F(s) defined by (1).

Various particular choices of the kernel function K(s, t) in (1) have led to a special transformation, each with its own properties to make it useful in specific circumstances. The transform defined choosing the kernel

(2)

is called Laplace transform, which is the one to which this book is devoted.

1.2 DEFINITION OF LAPLACE TRANSFORM

Let f(t) be a function for t > 0. The Laplace transform of f(t), denoted by L{f(t), is defined by

(3)

1.3 NOTATION

The integral in (3) is a function of the parameter s that is called F(s). It is customary to denote the functions of t by the lower case letters f, g, h, k, y, etc., and their Laplace transforms by the corresponding capital letters. Also some texts use capital letter L and show the Laplace transform of f(t) by L{f(t)}. Therefore we may write

(4)

If f(x) is a function of x e9780738672458_img_8807.gif 0, then its Laplace transform is denoted by L{f(x)}.

Example

Find L{sin at}.

By definition

By employing integration by parts

Therefore

Since for positive s, e−st → 0 at t → ∞, and sin at and cos at are bounded functions, therefore the above yields

1.4 LAPLACE TRANSFORMATION OF ELEMENTARY FUNCTIONS

The following table shows Laplace transforms of some elementary functions.

Note: Factorial n. For every integer n > 0,

factorial n = n! = 1 · 2 · ... · n

and by definition 0 ! = 1.

Table 1

1.5 SECTIONALLY OR PIECEWISE CONTINUOUS FUNCTIONS

The function f(t) is said to be piecewise continuous or sectionally continuous over an interval a < t < b if that interval can be divided into a finite number of intervals c < t < d such that

f(t) is continuous in the open interval c < t < d,

f(t) approaches a finite limit as t approaches each end point within the interval c < t < d; that is, the limits

exist and are finite, where ε > 0.

An example of a sectionally continuous function is shown in the following Figure 1.1.

Figure 1.1

1.6 FUNCTIONS OF EXPONENTIAL ORDER

The function f(t) is said to be of exponential order as t → ∞ if there exists two constants M > 0 and b and a fixed t0 such that

(5)

We also say f(t) is of exponential order b as