Module

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The Laplace Transform

Chapter 12  Fourier Series and the Laplace Transform

12.5  The Laplace Transform

In this section we investigate the Laplace transform, which is a very powerful tool for engineering applications.  It's discovery is attributed to the French mathematician Pierre-Simon Laplace (1749-1827).  The background we introduced in Section 12.4 regarding the Fourier transform in important for our approach to the theory of the Laplace transform.

12.5.1.  From the Fourier Transform to the Laplace Transform

We have shown that certain real-valued functions have a Fourier transform and that the integral

.

defines the complex function of the real variable .  If we multiply the integrand    by  ,  then we create a complex function    of the complex variable :

.

The function    is called the two-sided Laplace transform of , (or bilateral Laplace transform of ), and it exists when the Fourier transform of the function    exists.  From Fourier transform theory, a sufficient condition for    to exist is that

.

For a function , this integral is finite for values of that lie in some interval  .

The two-sided Laplace transform has the lower limit of integration and hence requires a knowledge of the past history of the function (i.e., when  ).  For most physical applications, we are interested in the behavior of a system only for .  The initial conditions    are a consequence of the past history of the system and are often all that we know.  For this reason, it is useful to define the one-sided Laplace transform of , which is commonly referred to simply as the Laplace transform of , which is also defined as an integral:

(12.28)            ,

where  .  If the integral in Equation (12.28) for the Laplace transform exists for  ,  then values of with imply that and so that

,

from which it follows that exists for .  Therefore the Laplace transform is defined for all points s in the right half-plane .

Another way to view the relationship between the Fourier transform and the Laplace transform is to consider the function given by

Then the Fourier transform theory, in Section 12.4, shows that

,

and, because the integrand is zero for , we can write this equation as

.

Now use the change of variable   and hold    fixed.  We have    and  .  Then the new limits of integration are from    to  . The resulting equation is

.

Definition (Laplace Transform).  Therefore the Laplace transform is as the integral:

,

where  ,  and the inverse Laplace transform is given by:

(12.29)            .

12.5.2  Properties of the Laplace Transform

Although a function may be defined for all values of t, it's Laplace transform is not influenced by values of , when .  The Laplace transform of is actually defined for the function given in the last section by

A sufficient condition for the existence of the Laplace transform is that    does not grow too rapidly as  .  We say that the function     is of exponential order if there exists real constants    such that

holds for all  .

All functions in this chapter are assumed to be of exponential order.  Theorem 12.10 shows that the Laplace transform    exists for values of    in a domain that includes the right half-plane  .

Theorem 12.10  (Laplace Transform).  If    is of exponential order, then its Laplace Transform   exists and is given by

,

where  .  The defining integral for    exists at points    in the right half plane  .

Proof.

Remark 12.1. The domain of definition of the defining integral for the Laplace transform seems to be restricted to a half plane.  However, the resulting formula might have a domain much larger than this half plane.  Later we will show that is an analytic function of the complex variable s.  For most applications involving Laplace transforms that we present, the Laplace transforms are rational functions that take the form  ,  where and   are polynomials; in other important applications, the functions take the form  .

Theorem 12.11  (Linearity of the Laplace Transform).  Let    have Laplace transforms  ,  respectively.  If  a  and  b  are constants, then

.

Proof.

Theorem 12.12  (Uniqueness of the Laplace Transform).  Let    have Laplace transforms  ,  respectively.

If  ,  then  .

Proof.

Table 12.2 gives the Laplace transforms of some well-known functions, and Table 12.3 highlights some important properties of Laplace transforms.

Example 12.7.  Show that the Laplace transform of the step function given by

is  .

Solution.

Using the integral definition for , we obtain

Explore Solution 12.7.

Example 12.8.  Show that  ,  where a is a real constant.

Solution.

We actually show that the integral defining equals the formula for values of s with and that the extension to other values of s is inferred by our knowledge about the domain of a rational function.  Using straightforward integration techniques gives

Let   be fixed, or where .  Then, as is a negative real number, we have  ,  which implies that  ,  and use this expression in the preceding equation to obtain  .

Explore Solution 12.8.

We can use the property of linearity to find new Laplace transforms from known transforms.

Example 12.9.  Show that  .

Solution.

Because can be written as the linear combination  ,  we obtain

Explore Solution 12.9.

Integration by parts is also helpful in finding new Laplace transforms.

Example 12.10.   Show that  .

Solution.

Integration by parts yields

For values of s in the right half-plane , an argument similar to that in Example 12.8 shows that the limit approaches zero, establishing the result.

Explore Solution 12.10.

Extra Example 1.   Show that  .

Explore Solution for Extra Example 1.

Example 12.11.   Show that  .

Solution.

A direct approach using the definition is tedious.  Instead, let's assume that the complex constants are permitted and hence that the following Laplace transforms exist:

and  .

Recall that can be written as the linear combination .  Using the linearity of the Laplace transform, we have

Inverting the Laplace transform is usually accomplished with the aid of a table of known Laplace transforms and the technique of partial fraction expansion. Table 12.2 gives the Laplace transforms of some well-known functions, and Table 12.3 highlights some important properties of Laplace transforms.

Example 12.12.  Find the inverse Laplace transform  .

Solution.

Using linearity and lines 6 and 7 of Table 11.2, we obtain

Explore Solution 12.12.

The Bromwich Integral for inverting the Laplace Transform

We will now investigate explore formula (12.29) which can be used to compute the inverse Laplace transform.

Definition of the Inverse Laplace Transform.

(12.28)                  ,

where  .

If the integral in Equation (12.28) for the Laplace transform exists for  ,  then values of with imply that and thus

,

from which it follows that exists for  .

Therefore the Laplace transform    is defined for all points s in the right half-plane  .

For many practical purposes, the function    will have a Laplace transform    is defined at all points in the complex plane

except at a finite number of singular points     where    has poles.  This is the situation we will consider.

Definition of the Inverse Laplace Transform.  The inverse Laplace Transform is defined with a contour integral

(12.29)                  ,

the Bromwich contour    is a vertical line in the complex plane where all singularities of

lie in the left half-plane  .   This integral is called the Bromwich integral (and sometimes it is called the Fourier-Mellin integral).

The singularities    of    lie to the left of the Bromwich contour.

We can use the Residue Calculus to evaluate the Bromwich integral.  The details are left for the reader to investigate.

We shall assume that the singularities of lie inside the simple closed contour consisting of the portion of the Bromwich contour

and a semicircle    of radius  R.

The singularities    of    lie inside the contour  .

The Cauchy Residue Theorem can be used to evaluate the contour integral along

.

Then

.

Taking limits we have

.

If sufficient conditions are imposed on   then it can be shown that

.

In Section 12.9 we will investigate functions of the form  , where are polynomials of degree m and n,

respectively, and  .  This will insure that

.

For this case we can use the complex function      and write

This result will be formally stated in Section 12.9 as the following theorem.

Theorem (Inverse Laplace Transform).  Let , where are polynomials of degree  ,  respectively, and  .

The inverse Laplace transform of    can be computed using residues, and is given by

,

where the sum is taken over all the singularities    of  .

Extra Example 2.   Evaluate a Bromwich contour integral to find the inverse Laplace transform   .

Extra Solution 2.

Extra Example 3.   Evaluate a Bromwich contour integral to find the inverse Laplace transform   .

Extra Solution 3.

Table 12.2 Table of Laplace Transforms

Table 12.3 Properties of Laplace Transforms

Library Research Experience for Undergraduates

Fourier Series and Transform

Laplace Transform

The Next Module is

Laplace Transforms of Derivatives and Integrals

(c) 2012 John H. Mathews, Russell W. Howell