Exercises for Section 7.5.  Applications of Taylor and Laurent Series

Exercise 1.  Determine whether there exists a function    that is analytic at    such that for  ,  

1 (a).    and  .
Solution 1 (a).

1 (b).  .  
Solution 1 (b).

1 (c).  .  
Solution 1 (c).

Exercise 2.  Prove the following corollaries and theorem.  

2 (a).  Corollary 7.9.

2 (b).  Corollary 7.10.

2 (c).  Theorem 7.14.

2 (d).  Corollary 7.12.

Exercise 3.  Consider the function  .  

3 (a).  Show that there is a sequence    of points converging to    such that    for  ,  
Solution 3 (a).

3 (b).  Does this result contradict Corollary 7.9 ?   Why or why not ?
Solution 3 (b).

Exercise 4  Let  .   

4 (a).  Use Theorem 7.14 to find the first few terms of the Maclaurin series for  ,    if  .  

4 (b).  What are the values of    and   ?  

Exercise 5.  Show that the real function    defined by  

is continuous at  ,  but that the corresponding complex function    defined by

is not continuous at  .
Solution 5.

Exercise 6.  Use L'Hôpital's rule to find the following limits.  

6 (a).   .

6 (b).   .  

6 (c).   .  

6 (d).   .  

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