Exercises for Section 8.1.  The Residue Theorem

Exercise 1.  Find      for the following functions:

1 (a).   .
Solution 1 (a).

1 (b).   .
Solution 1 (b).

1 (c).   .
Solution 1 (c).

1 (d).   .
Solution 1 (d).

1 (e).   .
Solution 1 (e).

1 (f).   .
Solution 1 (f).

1 (g).   .
Solution 1 (g).

1 (h).   .
Solution 1 (h).

1 (i).   .
Solution 1 (i).

1 (j).   .
Solution 1 (j).

1 (k).   .
Solution 1 (k).

1 (l).   .
Solution 1 (l).

1 (m).   .
Solution 1 (m).

1 (n).   .
Solution 1 (n).

Exercise 2.  Let    have an isolated singularity at  .

Show that   .

Solution 2.

Exercise 3.  Evaluate

3 (a).  .
Solution 3 (a).

3 (b).  .
Solution 3 (b).

3 (c).  .
Solution 3 (c).

3 (d).  .
Solution 3 (d).

3 (e).  .
Solution 3 (e).

3 (f).  .
Solution 3 (f).

3 (g).  .
Solution 3 (g).

Exercise 4.  Let    be analytic at .

If    and    has a simple zero at  ,  then show that  .

Solution 4.

Exercise 5.  Find  ,   when

5 (a).  .
Solution 5 (a).

5 (b).  .
Solution 5 (b).

Exercise 6.  Find  ,   when

6 (a).  .
Solution 6 (a).

6 (b).  .
Solution 6 (b).

Exercise 7.  Find  ,   when

7 (a).  .
Solution 7 (a).

7 (b).
Solution 7 (b).

Exercise 8.  Find  ,   when

8 (a).  .
Solution 8 (a).

8 (b).  .
Solution 8 (b).

Exercise 9.  Use residues to find the partial fraction representations of

9 (a).  .
Solution 9 (a).

9 (b).  .
Solution 9 (b).

9 (c).  .
Solution 9 (c).

9 (d).  .
Solution 9 (d).

9 (e).  .
Solution 9 (e).

9 (f).  .
Solution 9 (f).

Exercise 10.  Let    be analytic in a simply connected domain  D,  and let  C  be a simple closed positively oriented contour in  D.

If    is the only zero of    in  D  and    lies interior to  C,  then show that

,   where  k  is the order of the zero at  .
Solution 10.

Exercise 11.  Let   be analytic at the points  .

If   ,   then show that      for  .

Hint:  The solution is a little tricky.  You might need to use  .

Solution 11.

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