Example 8.13.   [Graphics:Images/IntegralsRationalMod_gr_35.gif]   does not exist,     and     [Graphics:Images/IntegralsRationalMod_gr_36.gif].   

                    [Graphics:Images/IntegralsRationalMod_gr_37.gif]  

Explore Solution 8.13.

Solution.  

        If we attempt to use Equation (8-7) then we obtain

                    [Graphics:../Images/IntegralsRationalMod_gr_45.gif]  

and the last computation   "[Graphics:../Images/IntegralsRationalMod_gr_46.gif]"   is undefined .  

Thus, the improper integral   [Graphics:../Images/IntegralsRationalMod_gr_47.gif]   does not exist.  

        If we use Equation (8-8) then we obtain

                    [Graphics:../Images/IntegralsRationalMod_gr_48.gif]   

This computation is well defined and is known as the Cauchy principal value ([Graphics:../Images/IntegralsRationalMod_gr_49.gif]) of   [Graphics:../Images/IntegralsRationalMod_gr_50.gif].   

Therefore,

                    [Graphics:../Images/IntegralsRationalMod_gr_51.gif].    

 

                    [Graphics:../Images/IntegralsRationalMod_gr_52.gif]

                         The area   [Graphics:../Images/IntegralsRationalMod_gr_53.gif]   "cancels out" the area   [Graphics:../Images/IntegralsRationalMod_gr_54.gif].  

                         Here we can see the value of the integrals, and that    [Graphics:../Images/IntegralsRationalMod_gr_55.gif].

 

We are done.   

 

Aside.  Both [Graphics:../Images/IntegralsRationalMod_gr_56.gif] and [Graphics:../Images/IntegralsRationalMod_gr_57.gif] can be used to investigate the integrals.

Aside.  We can let Mathematica compute the improper integral.

If we attempt to use Equation (8-7) then we obtain

[Graphics:../Images/IntegralsRationalMod_gr_58.gif]

[Graphics:../Images/IntegralsRationalMod_gr_59.gif]

If we use Equation (8-8) then we obtain

[Graphics:../Images/IntegralsRationalMod_gr_60.gif]

[Graphics:../Images/IntegralsRationalMod_gr_61.gif]


[Graphics:../Images/IntegralsRationalMod_gr_62.gif]

[Graphics:../Images/IntegralsRationalMod_gr_63.gif]


[Graphics:../Images/IntegralsRationalMod_gr_64.gif]

[Graphics:../Images/IntegralsRationalMod_gr_65.gif]

Aside.  Mathematica Vers. 7 and Vers. 8 can find the Principal Value of the integral.

[Graphics:../Images/IntegralsRationalMod_gr_66.gif]

[Graphics:../Images/IntegralsRationalMod_gr_67.gif]

We are really done.   

 

Aside.  We can let Maple compute the improper integral.

If we attempt to use Equation (8-7) then we obtain

     > [Graphics:../Images/IntegralsRationalMod_gr_68.gif]

               
[Graphics:../Images/IntegralsRationalMod_gr_69.gif]

If we use Equation (8-8) then we obtain

     > [Graphics:../Images/IntegralsRationalMod_gr_70.gif]

               
[Graphics:../Images/IntegralsRationalMod_gr_71.gif]

     > [Graphics:../Images/IntegralsRationalMod_gr_72.gif]

               
[Graphics:../Images/IntegralsRationalMod_gr_73.gif]

     > [Graphics:../Images/IntegralsRationalMod_gr_74.gif]

               
[Graphics:../Images/IntegralsRationalMod_gr_75.gif]

Aside.  Maple 12 can find the Cauchy Principal Value of the integral.

     > [Graphics:../Images/IntegralsRationalMod_gr_76.gif]

               
[Graphics:../Images/IntegralsRationalMod_gr_77.gif]

Remark.  In this book the use of computers is optional.  

Hopefully this text will promote their use and understanding.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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