ABSTRACT
Example 10.1. Regions of the Complex Plane. Determine the region of the complex z-plane specified: (I) by |z−a| > |z− b| for real
b > a; (II) by |z| ≥ a≥ |z+a| for real a, and find its angular points. Concerning (I) the meaning of |z−a|> |z− b| is that z is farther from a than from b, so
it must lie (Figure 10.1a) to the left of the bisector or the segment (a,b), viz.:
b > a : |z−a|> |z− b| ⇔Re(z)> a+ b 2
. (10.1)
Concerning (II) the point z must lie inside (outside) the circle of radius a and center at (−a,0) [the origin], hence it lies in a crescent shaped region (Figure 10.1b), which has two angular points z±; these are at distance a from the origin and the point (a,0), and thus define equilateral triangles, with sides a and internal angles π/3, so:
z± = a exp ( ± iπ
) =
a
( −1± i
√ 3 ) , (10.2)
are the locations of the angular points.
