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
Laplace transform of ),
and it exists when the Fourier transform of the
function exists. From
Fourier transform theory, a sufficient condition
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:
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
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
the Laplace transform is as the
where , and the inverse Laplace transform is given by:
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 .
Transform). If is
of exponential order, then its Laplace
exists and is given by
where . The defining integral for exists at points in the right half plane .
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
Transform). Let have
Laplace transforms , respectively. If a and b are
Theorem 12.12 (Uniqueness of
Transform). Let have
Laplace transforms , respectively.
If , then .
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
Using the integral definition for
Explore Solution 12.7.
Example 12.8. Show that , where a is a real constant.
We actually show that the integral
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
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 .
can be written as the linear combination , we
Explore Solution 12.9.
Integration by parts is also helpful in finding new Laplace transforms.
Example 12.10. Show that .
Integration by parts
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 .
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
Recall that can be written as the linear combination . Using the linearity of the Laplace transform, we have
Explore Solution 12.11.
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 .
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
will now investigate explore formula
(12.29) which can be used to compute the
inverse Laplace transform.
Definition of the Inverse Laplace Transform.
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
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
Taking limits we have
If sufficient conditions are imposed on then it can be shown that
12.9 we will investigate functions of the
are polynomials of degree m and
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 ,
are polynomials of degree , respectively,
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
Exercises for Section 12.5. The Laplace Transform
Fourier Series and Transform
The Next Module is
Laplace Transforms of Derivatives and Integrals
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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell