Exercises for Section 2.5.  The Reciprocal Transformation  

      For Exercises 1-8, find the image of the given circle or line under the reciprocal transformation  .

Hint.  The inverse mapping is  .   

Use the substitution  .

Exercise 1.  Find the image of the horizontal line   under the reciprocal transformation  .  

Make sketches and indicate the points    in the z-plane and their images    in the w-plane.
Solution 1.

Exercise 2.  Find the image of the circle    under the reciprocal transformation  .  

Make sketches and indicate the points   in the z-plane and their images in the w-plane.
Solution 2.

Exercise 3.  Find the image of the vertical line    under the reciprocal transformation  .  

Make sketches and indicate the points    in the z-plane and their images in the w-plane.
Solution 3.

Exercise 4.  Find the image of the circle    under the reciprocal transformation  .  

Make sketches and indicate the points   in the z-plane and their images in the w-plane.
Solution 4.

Exercise 5.  Find the image of the line    under the reciprocal transformation  .  

Make sketches and indicate the points   in the z-plane and their images in the w-plane.
Solution 5.

Exercise 6.  Find the image of the circle    under the reciprocal transformation  .  

Make sketches and indicate the points    in the z-plane and their images in the w-plane.
Solution 6.

Exercise 7.  Find the image of the circle     under the reciprocal transformation  .  

Make sketches and indicate the points   in the z-plane and their images in the w-plane.
Solution 7.

Exercise 8.  Find the image of the circle    under the reciprocal transformation  .  

Make sketches and indicate the points  ,   in the z-plane and their images in the w-plane.
Solution 8.

Exercise 9.  Limits involving  .  

The function     is said to have the limit  L  as  z  approaches  ,  and we write    iff
for every    there exists an    such that    (i.e., )  whenever  .  

Likewise,    iff   for every    there exists    such that  
  whenever    (i.e., ).

Use this definition to  

9 (a).  Show that  .
Solution 9 (a).

9 (b).  Show that  .  
Solution 9 (b).

Exercise 10.  Show that the reciprocal transformation    maps the vertical strip    onto the region in the right half-plane    that lies outside the unit circle  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images in the w-plane.
Solution 10.

Exercise 11.  Find the image of the disk    under  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images in the w-plane.
Solution 11.

Exercise 12.  Show that the reciprocal transformation maps the disk    onto the region that lies exterior to the circle  ,  i. e. the image region is  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images    in the w-plane.
Solution 12.

Exercise 13.  Find the image of the half-plane    under the mapping  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images    in the w-plane.
Solution 13.

Exercise 14.  Show that the half-plane    is mapped onto the disk    by the reciprocal transformation.  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images    in the w-plane.
Solution 14.

Exercise 15.  Find the image of the quadrant    under the mapping  .  

Make sketches of the domain set and range set.

Hint.  Consider the points  ,  ,  ,  ,    in the z-plane
and their images  ,  ,  ,  ,    in the w-plane.
Solution 15.

Exercise 16.  Show that the transformation    maps the disk    onto the lower half-plane  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images   in the w-plane.
Solution 16.

Exercise 17.  Show that the transformation    maps the disk    onto the right half-plane  .  

Make sketches of the domain set and range set.

Hint.  Consider the points    in the z-plane and their images   in the w-plane.
Solution 17.

Exercise 18.  Show that the parabola    is mapped onto the cardioid    by the reciprocal transformation.  
Solution 18.

Exercise 19.  Use the definition in Exercise 9 to prove that  .  
Solution 19.

Exercise 20.  Show that    is mapped onto the point    on the Riemann sphere.  
Solution 20.

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

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