17.3 Sets in the Complex Plane

INTRODUCTION

In the preceding sections we examined some rudiments of the algebra and geometry of complex numbers. But we have barely scratched the surface of the subject known as complex analysis; the main thrust of our study lies ahead. Our goal in the sections and chapters that follow is to examine functions of a single complex variable z = x + iy and the calculus of these functions.

Before introducing the notion of a function of a complex variable, we need to state some essential definitions and terminology about sets in the complex plane.

Terminology

Before discussing the concept of functions of a complex variable, we need to introduce some essential terminology about sets in the complex plane.

Suppose z0 = x0 + iy0. Since |z − z0| = is the distance between the points z = x + iy and z0 = x0 + iy0, the points z = x + iy that satisfy the equation

ρ > 0, lie on a circle of radius ρ centered at the point z0. See FIGURE 17.3.1.

An image shows a circle labeled mod(z minus z subscript 0) = rho. The center of the circle is labeled z subscript 0 and the radius of the circle is labeled rho.

FIGURE 17.3.1 Circle of radius ρ

EXAMPLE 1 Circles

(a) |z| = 1 is the equation of a unit circle centered at the origin.

(b) |z − 1 − 2i| = 5 is the equation of a circle of radius 5 centered at 1 + 2i. ≡

The points z satisfying the inequality z − z0 < ρ, ρ > 0, lie within, but not on, a circle of radius ρ centered at the point z0. This set is called a neighborhood of z0 or an open disk. A point z0 is said to be an interior point of a set S of the complex plane if there exists some neighborhood of z0 that lies entirely within S. If every point z of a set S is an interior point, then S is said to be an open set. See FIGURE 17.3.2. For example, the inequality Re(z) > 1 defines a right half-plane, which is an open set. All complex numbers z = x + iy for which x > 1 are in this set. If we choose, for example, z0 = 1.1 + 2i, then a neighborhood of z0 lying entirely in the set is defined by |z − (1.1 + 2i)| < 0.05. See FIGURE 17.3.3. On the other hand, the set S of points in the complex plane defined by Re(z) ≥ 1 is not open, since every neighborhood of a point on the line x = 1 must contain points in S and points not in S. See FIGURE 17.3.4.

An image has two diagonally opposite points. Each point is enclosed in a dashed circle. The point on the bottom left is labeled z subscript 0. Both the points and their respective circles are within in a shaded area.

FIGURE 17.3.2 Open set

A graph. The horizontal axis is labeled x and the vertical axis is labeled y. A vertical line close to the vertical axis and intersecting the horizontal axis on the left is labeled x = 1. The graph shows two opposite points. The point on the left is almost on the vertical line and is encircled by a small circle. The point on the right is encircled within a bigger circle and is labeled mod z minus (1.1 plus 2 i) is less than 0.05 and z = 1.1 plus 2 i. Two dashed tangential lines from the smaller circle and a dashed vertical line in the right of the bigger circle form a triangle.

FIGURE 17.3.3 Open set magnified view of a point near x = 1

A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a vertical line close to the vertical axis and intersecting the horizontal axis labeled x = 1. The area on the right of the vertical line is shaded. The graph also shows a dot in the center of the vertical line. The dot is enclosed within a dashed circle, which has other smaller dots inside it. Three smaller dots on the left are labeled not in S and two smaller dots on the bottom right are labeled in S.

FIGURE 17.3.4 Set S is not open

EXAMPLE 2 Open Sets

FIGURE 17.3.5 illustrates some additional open sets.

Four graphs. The horizontal axis is labeled x and the vertical axis is labeled y. In the first graph, the third and the fourth quadrants of the graph are shaded and titled Im (z) is less than 0, lower half-plane. The second graph shows two dashed vertical lines intersecting the horizontal axis on the right and left of the vertical axis. The area bound by the vertical lines is shaded and titled minus 1 is less than Re (z) is less than 1, infinite strip. The third graph shows a circle centered at the origin. The area outside the circle is shaded and titled mod(z) is greater than 1, exterior of unit circle. The fourth graph shows two concentric circles centered at the origin. The area of the circular ring formed by the circles is shaded and titled 1 is less than mod(z) is less than 2, circular ring.

FIGURE 17.3.5 Four examples of open sets ≡

The set of numbers satisfying the inequality

such as illustrated in Figure 17.3.5(d), is called an open annulus.

If every neighborhood of a point of a set S contains at least one point of S and at least one point not in S, then is said to be a boundary point of S. For the set of points defined by the points on the vertical line are boundary points. The points that lie on the circle are boundary points for the disk , as well as for the neighborhood of The collection of boundary points of a set S is called the boundary of S. The circle is the boundary for both the disk and the neighborhood of A point that is neither an interior point nor a boundary point of a set S is said to be an exterior point of S; in other words, is an exterior point of a set S if there exists some neighborhood of that contains no points of S. FIGURE 17.3.6 shows a typical set S with interior, boundary, and exterior.

An illustration. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a closed curve, which loosely resembles a fish. The tail of the curve touches the vertical axis and is labeled Interior. The head of the fish is labeled S. The outer part of the curve is labeled Exterior and its circumference is labeled Boundary.

FIGURE 17.3.6 Interior, boundary, and exterior of set S

If any pair of points z1 and z2 in an open set S can be connected by a polygonal line that lies entirely in the set, then the open set S is said to be connected. See FIGURE 17.3.7. An open connected set is called a domain. All the open sets in Figure 17.3.5 are connected and so are domains. The set of numbers satisfying Re(z) ≠ 4 is an open set but is not connected, since it is not possible to join points on either side of the vertical line x = 4 by a polygonal line without leaving the set (bear in mind that the points on x = 4 are not in the set).

An image with a U-shaped curve that starts at a point, goes down then goes to the right and then goes upward and ends at a point. It is enclosed within a shaded area. The right part of the curve is extended. The curve has end points labeled z subscript 1 at the left and z subscript 2 at the right.

FIGURE 17.3.7 Connected set

A region is a domain in the complex plane with all, some, or none of its boundary points. Since an open connected set does not contain any boundary points, it is automatically a region. A region containing all its boundary points is said to be closed. The disk defined by |z − i| ≤ 2 is an example of a closed region and is referred to as a closed disk. A region may be neither open nor closed; the annular region defined by 1 ≤ |z − 5| < 3 contains only some of its boundary points and so is neither open nor closed.

REMARKS

Often in mathematics the same word is used in entirely different contexts. Do not confuse the concept of “domain” defined in this section with the concept of the “domain of a function.”

17.3 Exercises Answers to selected odd-numbered problems begin on page ANS-45.

In Problems 1–8, sketch the graph of the given equation.

  1. Re(z) = 5
  2. Im(z) = −2
  3. Im( + 3i) = 6
  4. Im(z − i) = Re(z + 4 − 3i)
  5. |z − 3i| = 2
  6. |2z + 1| = 4
  7. |z − 4 + 3i| = 5
  8. |z + 2 + 2i| = 2

In Problems 9–22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain.

  1. Re(z) < −1
  2. |Re(z)| > 2
  3. Im(z) > 3
  4. Im(z − i) < 5
  5. 2 < Re(z − 1) < 4
  6. −1 ≤ Im(z) < 4
  7. Re(z2) > 0
  8. Im(1/z) <
  9. 0 ≤ arg (z) ≤ 2π/3
  10. |arg (z)| < π/4
  11. |z − i| > 1
  12. |z − i| > 0
  13. 2 < |z − i| < 3
  14. 1 ≤ |z − 1 − i| < 2
  15. Describe the set of points in the complex plane that satisfies |z + 1| = |z − i|.
  16. Describe the set of points in the complex plane that satisfies |Re(z)| ≤ |z|.
  17. Describe the set of points in the complex plane that satisfies z2 + = 2.
  18. Describe the set of points in the complex plane that satisfies |z − i| + |z + i| = 1.