Exercise 3.  Show that    is not a bounded function.

Solution 3.

See text and/or instructor's solution manual.

Solution.

We know that the complex cosine is an entire function that is not a constant.  

Therefore, by Theorem 6.18 (Liouville's Theorem),    is not bounded.

We are done.

Alternate solution.  We established this characteristic for    with a somewhat tedious proof in Section 5.4.  

Using Identity  (5-35) gives   .  

               

The identities    and    then yield  

(5-37)            .  

If we set in Identity (5-37) and let  ,  we get  

            

As advertised, we have shown that    is not a bounded function.

We are really done.

Aside.  We can graph  ,  this is just for fun.

The mapping    will map the infinite strip onto the w-plane.

This solution is complements of the authors.

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