Problem 20 - 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. $$ |z-i|>0 $$

Step-by-Step Solution

Answer

The set is the entire complex plane except the point \((0,1)\), and it is a domain.

Step 1: Understand the Inequality

The inequality states \(|z-i| > 0\). Here, \(z\) is a complex number represented as \(z = x + yi\), where \(x\) and \(y\) are real numbers. The term \(i\) is the imaginary unit which represents the point \(0 + 1i\) on the complex plane. The given inequality asks for all points such that the distance from \(z\) to \(i\) is greater than zero.

Step 2: Interpret the Distance Condition

The distance \(|z-i|\) between two complex numbers \(z = x + yi\) and \(i = 0 + 1i\) is given by the modulus \(|z-i| = \sqrt{x^2 + (y-1)^2}\). The inequality \(|z-i| > 0\) is satisfied for all points \(z\) except the point where \(z = i\), meaning \(x = 0\) and \(y = 1\).

Step 3: Sketch the Inequality on the Complex Plane

To sketch the points satisfying \(|z-i|>0\), plot the point \((0, 1)\) on the complex plane. All points in the complex plane except this one represent a distance from \(i\) that is greater than zero. Therefore, the solution includes the entire complex plane, excluding the point \((0, 1)\).

Step 4: Determine if the Set is a Domain

A domain in the complex plane is an open, connected set. Since the set includes all points except one, and the complex plane minus a single point is still open and connected, the set is indeed a domain.

Key Concepts

Complex Plane

In complex analysis, the complex plane is a vital concept that represents the entire set of complex numbers visually. It is akin to a 2D coordinate system where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part.
To understand this idea:

• Real numbers are plotted along the horizontal axis.
• Imaginary numbers are plotted along the vertical axis.
• Every point in the plane corresponds to a complex number in the form of \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.

Using the complex plane allows us to visualize arithmetic and other operations on complex numbers, such as addition and multiplication, in a geometric way. For example, with our exercise, we identified that a complex number \(z\) is located in the complex plane with respect to its position to the imaginary unit \(i\). Understanding the complex plane lays the groundwork for grasping more advanced topics in complex analysis.

Inequalities

Inequalities in the context of complex numbers are expressions that describe the relative position or size of complex numbers or sets in the complex plane. They tell us something about the "space" between numbers or how one number compares with another.
In our example, the inequality \(|z-i| > 0\) describes a condition:

• This condition signifies all points \(z\) in the complex plane whose distance from the point \(i\) is greater than zero.
• Graphically, this inequality represents the entire complex plane except for the single point \(i\).

Understanding inequalities like these helps identify regions or sets within the complex plane that meet specific conditions or behaviors. They are useful in defining boundaries and constraints in complex analysis problems.

Modulus of Complex Number

The modulus of a complex number, symbolized as \(|z|\), is a measure of its magnitude or size. It represents the distance from the origin \(0, 0\) in the complex plane to the point \(x, y\) representing the complex number \(z = x + yi\).
The formula to calculate the modulus is: \[|z| = \sqrt{x^2 + y^2}\]In our specific exercise, the modulus comes into play when determining the distance of \(z\) from \(i\). For \(|z-i|\), it is calculated as:\[|z-i| = \sqrt{x^2 + (y-1)^2}\]

• The modulus is always a non-negative value.
• If the modulus of a point from \(i\) is greater than zero, the point is not equal to \(i\).

Recognizing and calculating the modulus is crucial for determining relationships and distances in the complex plane, as it gives a tangible sense of how close or far apart two complex numbers really are.

Domain in Complex Analysis

A domain in complex analysis is a specific type of set in the complex plane, described by conditions that ensure certain properties are met. To be considered a domain, a set must be open and connected.
In our exercise, we needed to determine if the set satisfying \(|z-i|>0\) represents a domain. Let's break it down:

• **Open Set:** This means that every point within the set is an interior point. Removing a single point (in this case, \(i\)) from the complex plane does not affect its openness.
• **Connected Set:** This implies there is a path within the set between any two points in it without leaving the set. The entire complex plane (minus one point) remains connected because any point can reach another without stepping out of the set.

Therefore, the set described by \(|z-i|>0\), which includes all complex numbers except the point \(i\), still forms a domain. This concept helps in ensuring regions in complex analysis are suitable for differentiating and integrating complex functions, leading to more advanced topics such as holomorphic functions.