Problem 12 - 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. $$ \operatorname{Im}(z-i)<5 $$

Step-by-Step Solution

Answer

The set is the region below the line \( y = 6 \) in the complex plane; it is a domain.

Step 1: Understanding the Complex Number

A complex number is of the form \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. In our problem, \( z - i \) would shift the imaginary part, so the new imaginary part of \( z - i \) is \( y - 1 \).

Step 2: Substituting into the Inequality

The given inequality is \( \operatorname{Im}(z - i) < 5 \). Substitute \( z = x + yi \)n into the inequality, giving \( (y - 1) < 5 \).

Step 3: Solving the Inequality

Simplify the inequality \( y - 1 < 5 \) to find \( y < 6 \). This inequality describes the region where the imaginary part of \( z - i \) is less than 5, which corresponds to \( y < 6 \) for the original complex number \( z = x + yi \).

Step 4: Graphing the Inequality

Graph the inequality \( y < 6 \) in the complex plane. This represents the region below the horizontal line \( y = 6 \). The line itself is not included in the set.

Step 5: Determining if it is a Domain

A set is a domain if it is open and connected. Since \( y < 6 \) represents all points below the line \( y = 6 \) without including the line itself, the set is open. It is connected because it forms a continuous region below the line. Thus, the set is a domain.

Key Concepts

Complex Plane

The complex plane, often referred to as the Argand plane, is a two-dimensional plane where each point represents a complex number. Complex numbers are expressed in the form \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part. This visualization helps in understanding complex numbers by giving them a geometric interpretation.

In the complex plane, the horizontal axis is called the real axis, while the vertical axis is the imaginary axis. Therefore, a complex number \(z = 3 + 4i\) would appear as the point \((3, 4)\) where 3 units are along the real axis, and 4 units are along the imaginary axis.

This graphical representation aids in visualizing operations involving complex numbers, such as addition, subtraction, and more complex functions. For example, complex addition is simply vector addition in this plane. By treating complex numbers as vectors, we can understand how transformations affect them by observing changes in direction and magnitude. Graphical methods, like sketching inequalities, further enhance comprehension and allow students to solve complex analysis problems intuitively.

Inequalities

In the context of complex numbers, inequalities often involve comparing the real or imaginary parts of complex numbers. The inequality presented in the exercise, \(\text{Im}(z - i) < 5\), deals with the imaginary part of the complex number \(z\) after a transformation.

To solve an inequality involving complex numbers, it’s essential to break it down into more familiar components. Here, we know that if \(z = x + yi\), then \(z - i\) alters the imaginary part to \(y - 1\).

• The inequality \(y - 1 < 5\) simplifies to \(y < 6\), making it easier to interpret geometrically.
• This implies all complex numbers where the imaginary part (\(y\)) is less than 6 satisfy the inequality.
• Graphically, this demarcates a region on the complex plane, specifically below the line \(y = 6\).

Understanding how these inequalities shape regions helps in visualizing solutions and is fundamental in domains in complex analysis and related fields.

Domain in Complex Analysis

In complex analysis, a domain is a crucial concept defined as a subset of the complex plane that is open and connected. A set is open if it does not include its boundary, and connected if it forms a single, unbroken piece.

In the given exercise, the inequality \(y < 6\) describes an area below the line \(y = 6\) without including the line itself. This characteristic makes the set open, as the boundary (the line itself) is not part of the set.

• Being open means there are no boundary points included, allowing for every point in the region to have its neighborhood entirely within the domain.
• Connectedness ensures that any two points in the domain can be joined by a path lying entirely within it, confirming the region is unbroken.

Therefore, the set described by the inequality is a domain in the complex plane, satisfying these criteria of openness and connectedness, which are essential in complex analysis when studying functions and mappings in mathematical and applied contexts.