46 CHAPTER 1 • COMPLEX NUMBERS
- Sketch the curve z (t) = t^2 + 2t + i (t + 1)
(a) for -1 $. t $. 0.
(b) for 1 $. t $. 2.
Hint: Use x = t^2 + 2t, y = t + 1 and eliminate the para.meter t.
- Find a parametrization of the curve that is a portion of the parabola y = x^2 that
(a) joins the origin to the point 2 + 4i.
(b) joins the point -1 + i to the origin.
( c) joins the point 1 + i to the origin.- This exercise completes Example 1.26: Suppose that Re (Zo) > 0. Show that
Re (z) > 0 for aJI z E D, (zo), where e =Re (zo). - Find a parametrization of the curve that is a portion of the circle lzl = 1 that joins
the point - i to i if
(a) the curve is the right semicircle.
(b) the curve is the left semicircle. - Show that D 1 (0) is a domain and that 151 (0) = { z : lzl $. 1} is not a domain.
- Find a parametrization of the curve that is a portion of the circle C1 (0) that joins
the point 1 t-0 i if
(a) the parametrization is counterclockwise along the quarter circle.
(b) the parametrization is clockwise.- Fill in the details to complete Example 1.25. That is, show that
(a) the set {z: lzl > l} is the exterior of the set S.
(b) the set C1 (0) is tbe boundary of the set S.
9. Consider the following sets.
(i) {z: Re(z) > l}.
(ii) {z: - 1 <Im (z) $. 2}.
(iii) { z : lz - 2 -ii $. 2}.
(iv) {z: jz +3il > l}.
(v) {rei^9 :O<r<1 and -j <9< j}.
(vi) {re;^9 : r >I and i < (J < n.
(vii) {z: lzl < 1 or lz - 41 < 1 }.
(a) Sketch each set.
(b) State, with reasons, which of the following terms apply to the above
sets: open; connected; domain; region; closed region; bounded.1 0. Show that D 1 (0) is connected. Hint: Show that if z 1 and z2 lie in D 1 (0), then the
straight-line segment joining them lies entirely in D 1 (0).