Problem 17 - Advanced Engineering Mathematics

Question

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

It is a sector in the complex plane, but not a domain.

Step 1: Interpret the inequality

The inequality \(0 \leq \arg(z) \leq \frac{2\pi}{3}\) deals with the argument of a complex number \(z = x + yi\), which is the angle between the positive x-axis and the line segment joining the point \( (x,y) \) to the origin.

Step 2: Visualize the region in the complex plane

The argument \(\arg(z)\) describes a sector in the complex plane starting from the positive real axis (around 0 radians) extending to \(\frac{2\pi}{3}\) radians. This means the sector covers angles within the first quadrant and part of the second quadrant, essentially forming a slice of a circle centered at the origin.

Step 3: Boundary conditions

The boundaries of the set are where \(\arg(z) = 0\) and \(\arg(z) = \frac{2\pi}{3}\). These lines can be visualized from the origin: the first boundary lies along the positive real axis and the second boundary is arrayed at an angle of \(\frac{2\pi}{3}\) from it. Both boundaries are included due to the \(\leq\) signs.

Step 4: Determine if it is a domain

A domain in the complex plane is typically an open connected set that contains no isolated points. Although this set is connected, it is not open since its boundaries are part of the set. Therefore, it is not a domain.

Key Concepts

Argument of a Complex Number

The argument of a complex number, denoted as \( \arg(z) \), is a crucial concept in understanding the position of the number on the complex plane. A complex number \( z \) can be expressed in the form \( x + yi \), where \( x \) is the real part and \( yi \) is the imaginary part. The argument is essentially the angle measured from the positive x-axis to the line segment connecting the origin to the point \( (x, y) \). This angle helps in determining how far a complex number rotates clockwise or counterclockwise from the positive real x-axis.

• If \( \arg(z) = 0 \), the number lies on the positive real axis.
• If \( \arg(z) = \pi/2 \), the number is on the positive imaginary axis.
• The angle increases counterclockwise and decreases clockwise.

By considering angles that lie between \( 0 \) and \( 2\pi/3 \), we define a specific region in the complex plane, often visualized as a wedge or sector.

Inequalities in the Complex Plane

Inequalities involving arguments of complex numbers help us depict a specific region or set on the complex plane. An inequality such as \( 0 \leq \arg(z) \leq 2\pi/3 \) defines a sector of the plane. This sector starts from the positive x-axis and stretches to the line forming an angle of \( 2\pi/3 \) with it.
This includes points:

• Within the 'slice' formed from the origin to the boundary line at \( 2\pi/3 \).
• On both boundaries, because of the "\( \leq \)" symbols.

Visualizing these inequalities can guide us in sketching the region, offering a clearer understanding of complex number behaviors within defined constraints.

Complex Number Domains

A domain in the complex plane refers to an open and connected set. An open set is one where none of the boundary points are included, leading to the idea that you can move any small amount in any direction without leaving the set. However, the set described by \( 0 \leq \arg(z) \leq 2\pi/3 \) is not a domain because its boundaries are included as part of the set.

• The inclusion of points on \( \arg(z) = 0 \) and \( \arg(z) = 2\pi/3 \) makes the set closed along these lines.
• This means that while the set is fully connected, it does not satisfy the openness requirement.

Understanding the nature of domains helps us in complex analysis, particularly in defining functions that are analytic within these regions.