2.2  The Mappings and

    In Section 2.1 we studied linear mapping. Now we turn our attention to power functions.  For , we can express the function in polar coordinates by

            .  

If we also use polar coordinates for in the w -plane, we can express this mapping by the system of equations

            .  

    Because an argument of the product (z)(z) is twice an argument of z, we say that f doubles angles at the origin. Points that lie on the ray ,   are mapped onto points that lie on the ray   , .  If we now restrict the domain of to the region  

            ,  

then the image of A under the mapping can be described by the set  

            ,   

which consists of all points in the w-plane except the point w=0.

    The inverse mapping of f, which we shall denote by g, is then

            ,

where  .  That is

            ,

where  .  The function g is so important that we call special attention to it with a formal definition.

Definition 2.1 (Principal Square Root Function).  The function

            ,    for   .  

is called the principal square root function.

    It is left as an exercise to show that f and g satisfy equations g(f(z))=z and f(g(w))=w, and thus are inverses of each other that map the set A one-to-one and onto the set B and the set B one-to-one and onto the set A, respectively. Figure 2.12. illustrates this relationship.

        Figure 2.12  The mapping      (and  ).  

    What are the images of rectangles under the mapping ?  To find out, we make use of the Cartesian form  

            

and the resulting system of equations  

               

Example 2.12.  Show that the transformation  ,  usually maps vertical and horizontal lines onto parabolas and use this fact to find the image of the rectangle .  (a) Find the image of the vertical line  .  (b) Find the image of the horizontal line  .  

Solution.  Using the above equations, we determine that the vertical line    is mapped onto the set of points given by the equations    and  .  If  ,  then    and  

            ,  

this equation represents a parabola with vertex at  ,  oriented horizontally, and opening to the left.  If  ,  the set    has    precisely when  ,  so the part of the line    lying above the x axis is mapped to the top half of the parabola.  

    The horizontal line    is mapped onto the parabola given by the equations    and   .  If  ,  then as before we get  

            ,   

this equation represents a parabola with vertex at  ,  oriented horizontally and opening to the right.  If  ,  the part of the line    to the right of the y axis is mapped to the top half of the parabola because the set    has   precisely when  .

    Quadrant I is mapped onto quadrants I and II by  ,  so the rectangle    is mapped onto the region bounded by the top halves of the parabolas given by and and the u axis. The vertices    of the rectangle are mapped onto the four points ,  respectively, as indicated in Figure 2.13.

    Finally, we can easily verify that the vertical line    is mapped to the set .  This is simply the set of non-positive real numbers.  Similarly, the horizontal line    is mapped to the set , which is the set of non-negative real numbers.

Figure 2.13  The transformation  .

Explore Solution 2.12.

Extra Example 1. Consider the mapping  .  Find the image of the ray    and circle .  

Explore Extra Solution 1.

    We can use knowledge of the inverse mapping   to get further insight into how the mapping acts on rectangles.  If we let , then  

            ,  

and we note that the point    in the z plane is related to the point   in the w-plane by the system of equations  

                

Example 2.13. The transformation    usually maps vertical and horizontal lines onto portions of hyperbolas.  
(a)  Find the image of the vertical line  .   (b)  Find the image of the horizontal line  .  

Solution.  Let  .  The above equations map the right half-plane given by    (i.e.,  )  onto the region in the right half-plane satisfying    and lying to the right of the hyperbola  .  If  ,  the equations    map the upper half-plane    (i.e.,  )   onto the region in quadrant I satisfying    and lying above the hyperbola  . This situation is illustrated in Figure 2.14.  We leave as an exercise the investigation of what happens when  .  

            Figure 2.14  The mapping      and  ().  

Explore Solution 2.13.

    What happens to images of regions under the mapping  

               for  ,

where  ?  If we use polar coordinates for    in the w plane, we can represent this mapping by the system

            .  

This Equation indicate that the argument of f(z) is half the argument of z and that the modulus of f(z) is the square root of the modulus of z.  Points that lie on the ray    are mapped onto the ray  .  The image of the z plane (with the point z=0 deleted) consists of the right half-plane Re(w)>0 together with the positive v axis.  The mapping is shown in Figure 2.15.

        Figure 2.15  The mapping       (and  ).

Extra Example 2. Consider the mapping  .  Find the image of the ray    and circle .  

Explore Extra Solution 2.

    We can easily extend what we've done to integer powers greater than 2.  We begin by letting n be a positive integer, considering the function  ,  for  ,  and then expressing it in the polar coordinate form

            .  

If we use polar coordinates for in the w -plane, this mapping can be given by the system of equations  

              

    The image of the ray  ,   is the ray   ,   and the angles at the origin are increased by the factor n.  The functions    are periodic with period  ,  so f is in general an n-to-one function; that is, n points in the z-plane are mapped onto each non-zero point in the w-plane.

     If we now restrict the domain of    to the region

            ,  

then the image of E under the mapping    can be described by the set  

            ,   

which consists of all points in the w-plane except the point w=0. The inverse mapping of f, which we shall denote by g, is then

            ,

where .  That is
            ,

where .  As with the principle square root function, we make an analogous definition for   roots.

Definition 2.2 (Principal Root Function).  The function

            , for   

is called the principal   root function.

    We leave as an exercise to show that f and g are inverses of each other that map the set E one-to-one and onto the set F and the set F one-to-one and onto the set E, respectively. Figure 2.16 illustrates this relationship.

    Figure 2.16  The mapping      (and  ).  

Extra Example 3.  Explore the mapping  .  

Explore Extra Solution 3.

Extra Example 4.  Explore the mapping  .  

Explore Extra Solution 4.

Extra Example 5.  Explore the mapping  .  

Explore Extra Solution 5.

Exercises for Section 2.2.  The Mappings    and