Problem 20 - Advanced Engineering Mathematics

Question

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |z-i|>0 $$

Step-by-Step Solution

Answer

The set is the complex plane excluding the point \(i\) and is a domain.

Step 1: Understanding the Inequality

The inequality given is \(|z - i| > 0\). This can be interpreted as the set of complex numbers \(z\) whose distance from the point \(i\) on the complex plane is greater than zero.

Step 2: Interpreting |z-i|

The expression \(|z - i|\) represents the distance of a complex number \(z = x + yi\) from the point \(0 + i\) (which is \(i\)) on the imaginary axis. Therefore, \(|z - i| = \sqrt{(x-0)^2 + (y-1)^2}\).

Step 3: Analyzing the Inequality

The inequality \(|z - i| > 0\) tells us that \(z\) cannot be equal to \(i\). Therefore, the set includes all complex numbers except the point \(i\) itself.

Step 4: Set Description

The solution set includes every point in the complex plane except the single point \(i\). This can be described as the complex plane excluding one point.

Step 5: Determine if the Set is a Domain

A domain in complex analysis is an open connected set. This set is open because it excludes the boundary point \(i\), and it is connected because it's the entire complex plane minus one point, thus it remains as one piece. Hence, the set is indeed a domain.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to represent complex numbers visually. Each complex number corresponds to a unique point on this plane. A complex number is represented as \(z = x + yi\), where \(x\) is the real part, and \(y\) is the imaginary part of the number.

• The horizontal axis is known as the real axis, representing all possible values of the real part of complex numbers.
• The vertical axis is the imaginary axis, which represents all possible values of the imaginary part.

To visualize the point \(i\) on the complex plane, think of it as the point \(0 + i\), positioned at zero on the real axis and one on the imaginary axis. The complex plane helps provide a geometric insight into complex numbers and their operations.

Inequality

In the context of complex analysis, inequalities often refer to the distance between points on the complex plane. When we say \(|z - i| > 0\), we are establishing an inequality concerning distance. This particular inequality tells us that the distance between the complex number \(z\) and the point \(i\) is greater than zero.

• This means \(z\) cannot be exactly the same as \(i\) because the distance would then be zero, contradicting the inequality.
• Hence, every complex number except \(i\) satisfies this inequality.

Inequalities like these can help define regions in the complex plane by specifying a set of points that must satisfy certain conditions in relation to other points.

Distance

One of the fundamental concepts in the complex plane is the distance between two points. The expression \(|z - i|\) represents this distance. If \(z\) has coordinates \( (x, y) \) and \(i\) is at the position \( (0, 1) \) on the plane, the distance can be calculated using the formula:
\[|z - i| = \sqrt{(x-0)^2 + (y-1)^2}\]

• This formula is derived from the Pythagorean theorem, where the distance is the hypotenuse of a right triangle formed by \((x - 0)\) and \((y - 1)\).
• Graphically, this distance is represented as the straight line connecting point \(z\) to point \(i\).

Understanding how to compute this distance is crucial in complex analysis, as it is often used to define neighborhoods and open sets.

Domain

In complex analysis, a domain is a specific type of set on the complex plane that is open and connected. When we consider the set defined by the inequality \(|z - i| > 0\), we must determine if this set forms a domain.

• An open set doesn't include its boundary points, which in this case, means excluding the point \(i\).
• A connected set means there are no separate parts; the set forms a whole without disjointed pieces.

The set defined by \(|z - i| > 0\) is indeed a domain. It is open because it excludes \(i\) and remains connected because removing a single point from the complex plane does not disconnect it. Determining whether a set is a domain is important for many theorems and applications in complex analysis, as many functions are defined or analyzed based on their behavior over such domains.