Example 2.20.  We consider some branches of the two-valued square root function  ,  (where ).  Define the principal square root function as  

(2-28)            ,  

where    and   so that  .  The function is a branch of .  Using the same notation, we can find other branches of the square root function.  For example, if we let

(2-29)            ,    

    then
             

so can be thought of as "plus" and "minus" square root functions.  The negative real axis is called a branch cut for the functions .  Each point on the branch cut is a point of discontinuity for both functions .

Explore Solution 2.20.

Use Mathematica to graph the mapping   where .  

We can compare the above polar mapping with the cartesian coordinate mapping.

Use Mathematica to graph the mapping  .  

We can compare the above polar mapping with the cartesian coordinate mapping.

We have illustrated the two branches of the square root  .  

(c) 2006 John H. Mathews, Russell W. Howell