Exercises for Section 1.6.  The Topology of Complex Numbers

Exercise 1.  Find a parametrization of the line that  

1 (a).  joins the origin    to the point  .  
Solution 1 (a).

1 (b).  joins the point    to the point  .  
Solution 1 (b).

1 (c).  joins the point    to the point .  
Solution 1 (c).

1 (d).  joins the point    to the point  .  
Solution 1 (d).

Exercise 2.  Sketch the curve    

Hint: Use  ,  and     and eliminate the parameter  t.

2 (a).  for  .  
Solution 2 (a).

2 (b).  for  .
Solution 2 (b).

Exercise 3.  Find a parametrization of the curve that is a portion of the parabola    that  

3 (a).  joins the origin    to the point  .  
Solution 3 (a).

3 (b).  joins the point    to the origin  .  
Solution 3 (b).

3 (c).  joins the point    to the origin  .  
Solution 3 (c).

Exercise 4.  This exercise completes Example 1.26:   Suppose that  .  
Show that    for all  ,  where  .  
Solution 4.

Exercise 5.  Find a parametrization of the curve that is a portion of the circle    that joins the point    to    if  

5 (a).  the curve is the right semicircle.  
Solution 5 (a).

5 (b).  the curve is the left semicircle.  
Solution 5 (b).

Exercise 6.  Show that    is a domain and that    is not a domain.
Solution 6.

Exercise 7.  Find a parametrization of the curve that is a portion of the circle    that joins the point    to    if  

7 (a).  the parametrization is counterclockwise along the quarter circle.  
Solution 7 (a).

7 (b).  the parametrization is clockwise.  
Solution 7 (b).

Exercise 8.  Fill in the details to complete Example 1.25.  That is, show that

8 (a).  the set    is the exterior of the set      .
Solution 8 (a).

8 (b).  the set    is the boundary of the set  S.
Solution 8 (b).

Exercise 9.  Consider the following sets.   Sketch each set.   State, with reasons, which of the following terms apply to the above sets: open; connected; domain; region; closed region; bounded.  

9 (i).  .
Solution 9 (i).

9 (ii).  .
Solution 9 (ii).

9 (iii).  .
Solution 9 (iii).

9 (iv).  .
Solution 9 (iv).

9 (v).  .
Solution 9 (v).

9 (vi).  .
Solution 9 (vi).

9 (vii).  .  
Solution 9 (vii).

Exercise 10.  Show that    is connected.
Hint: Show that if   and   lie in  ,  then the straight-line segment joining them lies entirely in  .  
Solution 10.

Exercise 11.  Let    be a finite set of points.   Show that  S  is a bounded set.  
Solution 11.

Exercise 12.  Prove that the boundary of the neighborhood    is the circle  .  
Solution 12.

Exercise 13.  Let  S  be the open set consisting of all points z such that   or .   Show that  S  is not connected.  
Solution 13.

Exercise 14.  Prove that the only accumulation point of    is the point  0.  
Solution 14.

Exercise 15.  Regarding the relation between closed sets and accumulation points,  

15 (a).  Prove that if a set is closed, then it contains all its accumulations points.  
Solution 15 (a).

15 (b).  Prove that if a set  S  contains all its accumulation points, then  S  is closed.  
Solution 15 (b).

Exercise 16.  Prove that    is the set of accumulation points of  

16 (a).  The set  .   
Solution 16 (a).

16 (b).  The set  .   
Solution 16 (a).

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