Exercise 21.  Explain how the quantities  ,    and    differ.  How are they similar ?  

Solution 21.

See text and/or instructor's solution manual.

Solution.  Broadly speaking, the designations  ,    are designations for limits in Calculus indicating that a real quantity is becoming unbounded and positive or unbounded and negative, respectively.  Since complex numbers are not "positive or negative", there is no such designation in Complex Analysis.  

      However, the point    has been given a meaningful definition on the Riemann Sphere as the "north pole."  It is the limiting quantity when the modulus of an an arbitrary complex number  z  approaches  ,  and the complex number  z  is said to approach the point  .   One way this can happen is along a ray through the origin, i. e. if    is fixed, then  ,  and we can write  ,  where    is the point at infinity in the extended complex plane.

     In a loose sense the designations  ,    ,  can apply to real numbers on the x-axis that are embedded in the complex plane, i.e.

If  ,  then  .  

If  ,  then  .  

     But the modulus of a complex numbers can also approach    along the y-axis.

If  ,  then  .  

If  ,  then  .  

     In other words, we might really need to make the following definition

            .

We are done.   

Aside.  We can investigate the situation with Mathematica.

This solution is complements of the authors.

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