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 [Graphics:Images/LaplaceTransformMod_gr_1.gif] have a Fourier transform and that the integral


defines the complex function [Graphics:Images/LaplaceTransformMod_gr_3.gif] of the real variable [Graphics:Images/LaplaceTransformMod_gr_4.gif].  If we multiply the integrand  [Graphics:Images/LaplaceTransformMod_gr_5.gif]  by  [Graphics:Images/LaplaceTransformMod_gr_6.gif],  then we create a complex function  [Graphics:Images/LaplaceTransformMod_gr_7.gif]  of the complex variable [Graphics:Images/LaplaceTransformMod_gr_8.gif]:  


The function  [Graphics:Images/LaplaceTransformMod_gr_10.gif]  is called the two-sided Laplace transform of [Graphics:Images/LaplaceTransformMod_gr_11.gif], (or bilateral Laplace transform of [Graphics:Images/LaplaceTransformMod_gr_12.gif]), and it exists when the Fourier transform of the function  [Graphics:Images/LaplaceTransformMod_gr_13.gif]  exists.  From Fourier transform theory, a sufficient condition for  [Graphics:Images/LaplaceTransformMod_gr_14.gif]  to exist is that


    For a function [Graphics:Images/LaplaceTransformMod_gr_16.gif], this integral is finite for values of [Graphics:Images/LaplaceTransformMod_gr_17.gif] that lie in some interval  [Graphics:Images/LaplaceTransformMod_gr_18.gif].

    The two-sided Laplace transform has the lower limit of integration [Graphics:Images/LaplaceTransformMod_gr_19.gif] and hence requires a knowledge of the past history of the function [Graphics:Images/LaplaceTransformMod_gr_20.gif] (i.e., when  [Graphics:Images/LaplaceTransformMod_gr_21.gif]).  For most physical applications, we are interested in the behavior of a system only for [Graphics:Images/LaplaceTransformMod_gr_22.gif].  The initial conditions  [Graphics:Images/LaplaceTransformMod_gr_23.gif]  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 [Graphics:Images/LaplaceTransformMod_gr_24.gif], which is commonly referred to simply as the Laplace transform of [Graphics:Images/LaplaceTransformMod_gr_25.gif], which is also defined as an integral:

(12.28)            [Graphics:Images/LaplaceTransformMod_gr_26.gif],
where  [Graphics:Images/LaplaceTransformMod_gr_27.gif].  If the integral in Equation (12.28) for the Laplace transform exists for  [Graphics:Images/LaplaceTransformMod_gr_28.gif],  then values of [Graphics:Images/LaplaceTransformMod_gr_29.gif] with [Graphics:Images/LaplaceTransformMod_gr_30.gif] imply that [Graphics:Images/LaplaceTransformMod_gr_31.gif] and so that  


from which it follows that [Graphics:Images/LaplaceTransformMod_gr_33.gif] exists for [Graphics:Images/LaplaceTransformMod_gr_34.gif].  Therefore the Laplace transform [Graphics:Images/LaplaceTransformMod_gr_35.gif] is defined for all points s in the right half-plane [Graphics:Images/LaplaceTransformMod_gr_36.gif].

    Another way to view the relationship between the Fourier transform and the Laplace transform is to consider the function [Graphics:Images/LaplaceTransformMod_gr_37.gif] given by


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


and, because the integrand [Graphics:Images/LaplaceTransformMod_gr_40.gif] is zero for [Graphics:Images/LaplaceTransformMod_gr_41.gif], we can write this equation as


Now use the change of variable [Graphics:Images/LaplaceTransformMod_gr_43.gif]  and hold  [Graphics:Images/LaplaceTransformMod_gr_44.gif]  fixed.  We have  [Graphics:Images/LaplaceTransformMod_gr_45.gif]  and  [Graphics:Images/LaplaceTransformMod_gr_46.gif].  Then the new limits of integration are from  [Graphics:Images/LaplaceTransformMod_gr_47.gif]  to  [Graphics:Images/LaplaceTransformMod_gr_48.gif]. The resulting equation is



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


where  [Graphics:Images/LaplaceTransformMod_gr_51.gif],  and the inverse Laplace transform is given by:

(12.29)            .  


12.5.2  Properties of the Laplace Transform

    Although a function [Graphics:Images/LaplaceTransformMod_gr_53.gif] may be defined for all values of t, it's Laplace transform is not influenced by values of [Graphics:Images/LaplaceTransformMod_gr_54.gif], when [Graphics:Images/LaplaceTransformMod_gr_55.gif].  The Laplace transform of [Graphics:Images/LaplaceTransformMod_gr_56.gif] is actually defined for the function [Graphics:Images/LaplaceTransformMod_gr_57.gif] given in the last section by


A sufficient condition for the existence of the Laplace transform is that  [Graphics:Images/LaplaceTransformMod_gr_59.gif]  does not grow too rapidly as  [Graphics:Images/LaplaceTransformMod_gr_60.gif].  We say that the function   [Graphics:Images/LaplaceTransformMod_gr_61.gif]  is of exponential order if there exists real constants  [Graphics:Images/LaplaceTransformMod_gr_62.gif]  such that  

            [Graphics:Images/LaplaceTransformMod_gr_63.gif]  holds for all  [Graphics:Images/LaplaceTransformMod_gr_64.gif].  

All functions in this chapter are assumed to be of exponential order.  Theorem 12.10 shows that the Laplace transform  [Graphics:Images/LaplaceTransformMod_gr_65.gif]  exists for values of  [Graphics:Images/LaplaceTransformMod_gr_66.gif]  in a domain that includes the right half-plane  [Graphics:Images/LaplaceTransformMod_gr_67.gif].  


Theorem 12.10  (Laplace Transform).  If  [Graphics:Images/LaplaceTransformMod_gr_68.gif]  is of exponential order, then its Laplace Transform  [Graphics:Images/LaplaceTransformMod_gr_69.gif] exists and is given by


where  [Graphics:Images/LaplaceTransformMod_gr_71.gif].  The defining integral for  [Graphics:Images/LaplaceTransformMod_gr_72.gif]  exists at points  [Graphics:Images/LaplaceTransformMod_gr_73.gif]  in the right half plane  [Graphics:Images/LaplaceTransformMod_gr_74.gif].



Remark 12.1. The domain of definition of the defining integral for the Laplace transform [Graphics:Images/LaplaceTransformMod_gr_75.gif] seems to be restricted to a half plane.  However, the resulting formula [Graphics:Images/LaplaceTransformMod_gr_76.gif] might have a domain much larger than this half plane.  Later we will show that [Graphics:Images/LaplaceTransformMod_gr_77.gif] 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  [Graphics:Images/LaplaceTransformMod_gr_78.gif],  where [Graphics:Images/LaplaceTransformMod_gr_79.gif] and  [Graphics:Images/LaplaceTransformMod_gr_80.gif] are polynomials; in other important applications, the functions take the form  [Graphics:Images/LaplaceTransformMod_gr_81.gif].  


Theorem 12.11  (Linearity of the Laplace Transform).  Let  [Graphics:Images/LaplaceTransformMod_gr_82.gif]  have Laplace transforms  [Graphics:Images/LaplaceTransformMod_gr_83.gif],  respectively.  If  a  and  b  are constants, then   




Theorem 12.12  (Uniqueness of the Laplace Transform).  Let  [Graphics:Images/LaplaceTransformMod_gr_85.gif]  have Laplace transforms  [Graphics:Images/LaplaceTransformMod_gr_86.gif],  respectively.   

            If  [Graphics:Images/LaplaceTransformMod_gr_87.gif],  then  [Graphics:Images/LaplaceTransformMod_gr_88.gif].  



    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  [Graphics:Images/LaplaceTransformMod_gr_90.gif].  


    Using the integral definition for [Graphics:Images/LaplaceTransformMod_gr_91.gif], we obtain  


Explore Solution 12.7.


Example 12.8.  Show that  [Graphics:Images/LaplaceTransformMod_gr_99.gif],  where a is a real constant.


    We actually show that the integral defining [Graphics:Images/LaplaceTransformMod_gr_100.gif] equals the formula [Graphics:Images/LaplaceTransformMod_gr_101.gif] for values of s with [Graphics:Images/LaplaceTransformMod_gr_102.gif] 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 [Graphics:Images/LaplaceTransformMod_gr_104.gif]  be fixed, or where [Graphics:Images/LaplaceTransformMod_gr_105.gif].  Then, as [Graphics:Images/LaplaceTransformMod_gr_106.gif] is a negative real number, we have  [Graphics:Images/LaplaceTransformMod_gr_107.gif],  which implies that  [Graphics:Images/LaplaceTransformMod_gr_108.gif],  and use this expression in the preceding equation to obtain  [Graphics:Images/LaplaceTransformMod_gr_109.gif].

Explore Solution 12.8.


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


Example 12.9.  Show that  [Graphics:Images/LaplaceTransformMod_gr_121.gif].  


    Because [Graphics:Images/LaplaceTransformMod_gr_122.gif] can be written as the linear combination  [Graphics:Images/LaplaceTransformMod_gr_123.gif],  we obtain  


Explore Solution 12.9.


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


Example 12.10.   Show that  [Graphics:Images/LaplaceTransformMod_gr_144.gif].  


    Integration by parts yields  


For values of s in the right half-plane [Graphics:Images/LaplaceTransformMod_gr_146.gif], 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  [Graphics:Images/LaplaceTransformMod_gr_158.gif].  

Explore Solution for Extra Example 1.


Example 12.11.   Show that  [Graphics:Images/LaplaceTransformMod_gr_170.gif].  


    A direct approach using the definition is tedious.  Instead, let's assume that the complex constants [Graphics:Images/LaplaceTransformMod_gr_171.gif] are permitted and hence that the following Laplace transforms exist:

            [Graphics:Images/LaplaceTransformMod_gr_172.gif]  and  [Graphics:Images/LaplaceTransformMod_gr_173.gif].

Recall that [Graphics:Images/LaplaceTransformMod_gr_174.gif] can be written as the linear combination [Graphics:Images/LaplaceTransformMod_gr_175.gif].  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  [Graphics:Images/LaplaceTransformMod_gr_192.gif].  


    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)                  [Graphics:Images/LaplaceTransformMod.2_gr_1.gif],
where  [Graphics:Images/LaplaceTransformMod.2_gr_2.gif].  

        If the integral in Equation (12.28) for the Laplace transform exists for  [Graphics:Images/LaplaceTransformMod.2_gr_3.gif],  then values of [Graphics:Images/LaplaceTransformMod.2_gr_4.gif] with [Graphics:Images/LaplaceTransformMod.2_gr_5.gif] imply that [Graphics:Images/LaplaceTransformMod.2_gr_6.gif] and thus  


from which it follows that [Graphics:Images/LaplaceTransformMod.2_gr_8.gif] exists for  [Graphics:Images/LaplaceTransformMod.2_gr_9.gif].  

Therefore the Laplace transform  [Graphics:Images/LaplaceTransformMod.2_gr_10.gif]  is defined for all points s in the right half-plane  [Graphics:Images/LaplaceTransformMod.2_gr_11.gif].

        For many practical purposes, the function  [Graphics:Images/LaplaceTransformMod.2_gr_12.gif]  will have a Laplace transform  [Graphics:Images/LaplaceTransformMod.2_gr_13.gif]  is defined at all points in the complex plane
except at a finite number of singular points   [Graphics:Images/LaplaceTransformMod.2_gr_14.gif]  where  [Graphics:Images/LaplaceTransformMod.2_gr_15.gif]  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)                  [Graphics:Images/LaplaceTransformMod.2_gr_16.gif],   

the Bromwich contour  [Graphics:Images/LaplaceTransformMod.2_gr_17.gif]  is a vertical line in the complex plane where all singularities of  [Graphics:Images/LaplaceTransformMod.2_gr_18.gif]  

lie in the left half-plane  [Graphics:Images/LaplaceTransformMod.2_gr_19.gif].   This integral is called the Bromwich integral (and sometimes it is called the Fourier-Mellin integral).


                   The singularities  [Graphics:Images/LaplaceTransformMod.2_gr_21.gif]  of  [Graphics:Images/LaplaceTransformMod.2_gr_22.gif]  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 [Graphics:Images/LaplaceTransformMod.2_gr_23.gif] lie inside the simple closed contour consisting of the portion of the Bromwich contour

  [Graphics:Images/LaplaceTransformMod.2_gr_24.gif]  and a semicircle  [Graphics:Images/LaplaceTransformMod.2_gr_25.gif]  of radius  R.


                    The singularities  [Graphics:Images/LaplaceTransformMod.2_gr_27.gif]  of  [Graphics:Images/LaplaceTransformMod.2_gr_28.gif]  lie inside the contour  [Graphics:Images/LaplaceTransformMod.2_gr_29.gif].  

        The Cauchy Residue Theorem can be used to evaluate the contour integral along  [Graphics:Images/LaplaceTransformMod.2_gr_30.gif]    




Taking limits we have


If sufficient conditions are imposed on  [Graphics:Images/LaplaceTransformMod.2_gr_34.gif] then it can be shown that  


        In Section 12.9 we will investigate functions of the form  [Graphics:Images/LaplaceTransformMod.2_gr_36.gif], where [Graphics:Images/LaplaceTransformMod.2_gr_37.gif] are polynomials of degree m and n,

respectively, and  [Graphics:Images/LaplaceTransformMod.2_gr_38.gif].  This will insure that   


For this case we can use the complex function   [Graphics:Images/LaplaceTransformMod.2_gr_40.gif]   and write


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

Theorem (Inverse Laplace Transform).  Let [Graphics:Images/LaplaceTransformMod.2_gr_42.gif], where [Graphics:Images/LaplaceTransformMod.2_gr_43.gif] are polynomials of degree  [Graphics:Images/LaplaceTransformMod.2_gr_44.gif],  respectively, and  [Graphics:Images/LaplaceTransformMod.2_gr_45.gif].  

The inverse Laplace transform of  [Graphics:Images/LaplaceTransformMod.2_gr_46.gif]  can be computed using residues, and is given by  


where the sum is taken over all the singularities  [Graphics:Images/LaplaceTransformMod.2_gr_48.gif]  of  [Graphics:Images/LaplaceTransformMod.2_gr_49.gif].  

Extra Example 2.   Evaluate a Bromwich contour integral to find the inverse Laplace transform   [Graphics:Images/LaplaceTransformMod.2_gr_50.gif].  

Extra Solution 2.


Extra Example 3.   Evaluate a Bromwich contour integral to find the inverse Laplace transform   [Graphics:Images/LaplaceTransformMod.2_gr_83.gif].  

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  


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