Exercises for Section 10.4.  Mapping by Trigonometric Functions

Exercise 1.  Find the image of the semi-infinite strip      under the mapping   .  
Solution 1.

Exercise 2.  Find the image of the vertical strip    under the mapping   .
Solution 2.

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

Exercise 4.  Find the image of the horizontal line    under the transformation   .
Solution 4.

Exercise 5.  Find the image of the rectangle    under the transformation   .  
Solution 5.

Exercise 6.  Find the image of the semi-infinite strip    under the mapping   .
Solution 6.

Exercise 7 (a).  Find   .  
Solution 7 (a).

Exercise 7 (b).  Find   .  
Solution 7 (b).

Exercise 8.  Use formulas (10-26) and (10-27) to find the following:

Exercise 8 (a).  .
Solution 8 (a).

Exercise 8 (b).  .
Solution 8 (b).

Exercise 8 (c).  .
Solution 8 (c).

Exercise 8 (d).  .
Solution 8 (d).

Exercise 9.  Show that the function      maps the rectangle      one-to-one and onto

the portion of the upper half plane      that lies inside the ellipse   .
Solution 9.

Exercise 10.  Find the image of the vertical strip      under the mapping   .
Solution 10.

Exercise 11.  Find the image of the horizontal strip    under the mapping   .  
Solution 11.

Exercise 12.  Find the image of the right half plane    under the mapping   .  
Solution 12.

Exercise 13.  Find the image of the first quadrant    under the transformation   .  
Solution 13.

Exercise 14.  Find the image of the first quadrant    under the mapping   .
Solution 14.

Exercise 15.  Show that the transformation      is a one-to-one conformal mapping of

the semi-infinite strip      onto the upper half plane  .
Solution 15.

Exercise 16.  Find the image of the semi-infinite vertical strip      under the mapping   .
Solution 16.

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