Solution 13.

See text and/or instructor's solution manual.

Answer.   The inequality      in Jordan's Lemma would not be possible to get if we replaced    by either the complex    or  .  

Solution.

We can still use inequality (8-16)

(8-16)          ,  whenever  .  

Using the result of Exercise 16 (d) in Section 5.4 we have

                    .  

Suppose we replace    with complex    in the proof of Jordan's Lemma, (see Lemma 8.1) then inequality (8-16) would become:

                    
  
                      
  
                        
    
                        

      If we were able to complete this proof then we would need to show that the last quantity can be made less than  ,  but we have  

                    .

This can be parameterized as      which in turn can be written as  

                     .

      In the proof of Jordan's lemma we used the inequality      and were able to prove that  

                    ,    for    .

      This argument can still be used to show that the second term      can be made as "small as you wish."  

It is curious that Mathematica can compute the integral      analytically:

               The graph of   .

               We can see that   .

               This in turn shows that  ,    for    .

      However, there does not seem to be a way to make the term      "as small as you wish."  

Indeed, if      then   ,   and   ,   and  

          .  

Clearly,    as      and it cannot be made "as small as you wish."  

So there is no way to make      go to zero as   .

      This integral in question      can be handled by Mathematica

When we graph it, we see that it diverges to infinity and cannot be made "as small as you wish."  

               The graph of   .

               Hence,   ,    as    .

     This demonstration will suffice to conclude that an analytic proof cannot be constructed when we replace    by with complex  .  

Similarly, an analytic proof cannot be constructed when we replace    by with complex  .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell