2.2 • THE MAPPINGS W =Zn AND W = z!, 69
-------~EXERCISES FOR SECTION 2.2
- Find t he images of the mapping w = z^2 in each case, and sketch the mapping.
(a) The hor izontal line {(x, y): y = 1}.
(b) The vertical line {(x, y): x = 2 }.
(c) The rectangle {(x, y): 0 < x < 2, 0 < y < l}.
( d) The triangle with vertices 0, 2, and 2 + 2i.
(e) The infinite strip {(x, y): 1 < x < 2}.
(f) The right half-pla ne region to the right of the hyper bola x^2 - y^2 = 1.
(g) The first quadrant region between the hyperbolas xy = ~ a nd xy = 4.- For what values of z does ( z^2 ) ~ = z hold if t he principal value of the square root
is t o be used? - Sketch the set of points satisfying the following relations.
(a) Re (z^2 ) > 4.
(b) Im (z^2 ) > 6.- F ind a nd illustrate the images of the following sets u nder the mapping w = z'.
(a) { re^18 : r > 1 and i < 8 < ~}.
(b) { re^18 : 1 < r < 9 and 0 < 8 <^2 ;}.
(c) { re^18 : r < 4 and - ir < 8 < ~}.
(d) T he vertical line {(x, y): x = 4}.
(e) T he infinite strip { (x, y): 2 < y < 6}.
2
(£) T he region to the right of the parabola x = 4 - 1fi.
Hint: Use the inverse mapping z = w^2 to show that the answer is the
r ight half-plane Re ( w ) > 2.- Find t he image of the right half-plane Re(z) > 1 under the mappingw = z^2 +2z+l.
- F ind the image of t he following set s under t he mapping w = z^3.
(a) { re^19 : 1 < r < 2 and i < 8 < i}.
(b) { re'^9 : r > 3 and^2 ; < 8 <^3 ; }.- Find t he image of {rei^8 : r > 2, a nd i < 8 < i} u nder t he following mappings.
(a) w = z^3 •
(b) w = z^4 •