Problem 17 - Advanced Engineering Mathematics

Question

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ 0 \leq \arg (z) \leq 2 \pi / 3 $$

Step-by-Step Solution

Answer

The set is a sector between 0 and \(2\pi/3\) radians, and it is not a domain.

Step 1: Understanding the Inequality

The complex number inequality \(0 \leq \arg(z) \leq 2\pi/3\) specifies the set of complex numbers where the argument (or angle) of \(z\) is between 0 and \(2\pi/3\). Recall that the argument of a complex number represents its angle with the positive real axis, measured counterclockwise.

Step 2: Sketching the Region in the Complex Plane

To sketch the region, we plot angles 0 and \(2\pi/3\) on the complex plane. The angle 0 corresponds to the positive real axis, and \(2\pi/3\) is 120 degrees from there. This forms a sector of a circle, radiating from the origin outwards, covering an angle of \(2\pi/3\). Any point in this sector can be described as \(z = re^{i\theta}\) with \(0 \leq \theta \leq 2\pi/3\) and \(r \geq 0\).

Step 3: Verify if the Region is a Domain

A set is a domain if it is open and connected. Our set includes the boundary lines (rays from the origin at angles 0 and \(2\pi/3\), and any radial line segment between them), which makes the set closed, so it is not an open set. Hence, it is not a domain because a domain must be open.

Key Concepts

Complex Numbers

Complex numbers are numbers of the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). Complex numbers extend the idea of one-dimensional real numbers to the two-dimensional complex plane. This plane is similar to a Cartesian plane with the horizontal axis representing real parts and the vertical axis representing imaginary parts.

• The real part of \( z \) is \( a \).
• The imaginary part of \( z \) is \( b \).

Complex number operations such as addition and multiplication follow familiar rules but incorporate \( i \). For example, to add \((3 + 4i) + (1 + 2i) = (3+1) + (4+2)i = 4 + 6i\). In the complex plane, this is akin to vector addition.

Argument of Complex Numbers

The argument of a complex number is a measure of its angle in the complex plane relative to the positive real axis. For a complex number \( z = re^{i\theta} \), where \( r \) is the magnitude, \( \theta \) is the argument.

• The argument is typically in radians, between \( -\pi \) and \( \pi \), covering a full range around the circle.
• In polar form, \( z \,=\, r(\cos \theta + i \sin \theta)\), making it easy to understand angles related to the real axis.

In the given problem, the inequality \( 0 \leq \arg(z) \leq 2\pi/3 \) provides an angular range, showing us that feasible values for \( \theta \) lie between these limits. This restriction defines a sector within the plane.

Domains in Complex Analysis

In complex analysis, the concept of a domain is crucial for defining functions and analyzing their properties. A domain is an open, connected subset of the complex plane.

• Open sets exclude their boundary points.
• Connected sets mean there is a path within the set connecting any two points without leaving the set.

In the provided exercise, the set determined by \( 0 \leq \arg(z) \leq 2\pi/3 \) includes the boundaries at angles 0 and \( 2\pi/3 \), making it a closed set rather than open.
Since domains must be open, this set does not qualify as a domain, despite being connected.

Inequalities in Complex Plane

Inequalities in the complex plane often describe regions or areas where specific conditions are met by complex numbers. These inequalities are visual,
allowing us to understand and explore different properties within the context of complex numbers.

• We interpret the inequality \( 0 \leq \arg(z) \leq 2\pi/3 \) by considering the angles from the positive real axis.
• It defines a wedge-like section of the plane radiating from the origin.

The inequality ultimately helps visualize not just individual points but whole sections described by specific conditions. Such regions arecritical in understanding how functions behave and interact across different areas of the plane.
By exploring these inequalities, we can uncover valuable insights into continuity, boundaries, and limits in complex analysis.