Problem 13 - 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. $$ 2<\operatorname{Re}(z-1)<4 $$

Step-by-Step Solution

Answer

The set forms a vertical strip between lines \(x = 3\) and \(x = 5\), and it is a domain as it is open and connected.

Step 1: Understanding the Inequality

We are given the inequality \(2 < \operatorname{Re}(z-1) < 4\). The \text{Re} function returns the real part of a complex number, \(z\). Thus, the real part of \(z - 1\) is between 2 and 4.

Step 2: Expressing the Complex Variable

Let us express the complex number \(z\). If \(z = x + yi\) where \(x\) is the real part and \(yi\) is the imaginary part, then \(z - 1 = (x - 1) + yi\). This means that \(\operatorname{Re}(z - 1) = x - 1\).

Step 3: Rewriting the Inequality

Substitute \(\operatorname{Re}(z - 1) = x - 1\) into the inequality to get \(2 < x - 1 < 4\). Solving for \(x\), we add 1 to each part of the inequality to get \(3 < x < 5\).

Step 4: Visualizing on the Complex Plane

In the complex plane, this inequality describes a vertical strip or band where the real part of \(z\) ranges from 3 to 5. The imaginary part \(y\) can be any real number, so the strip extends infinitely in the vertical direction.

Step 5: Determine if the Set is a Domain

A domain in complex analysis is an open and connected set. The region described by \(3 < x < 5\) and any \(y\) is both open (it doesn't include the lines \(x = 3\) and \(x = 5\), only the values between them) and connected (all points in the strip are connected within the strip), so the set is indeed a domain.

Key Concepts

Complex Plane

The complex plane is like a playground for complex numbers. Each point on this plane represents a complex number. We express these numbers as \(z = x + yi\), where \(x\) is the real part and \(yi\) is the imaginary part.
To visualize, imagine a regular Cartesian coordinate system.

• The horizontal axis (x-axis) represents the real part of a complex number.
• The vertical axis (y-axis) represents the imaginary part.

This helps us plot complex numbers as points. To sketch or solve problems in complex analysis, understanding how to locate complex numbers on this plane is crucial.
In essence, just view the complex plane as a map for complex numbers, complete with latitude (imaginary part) and longitude (real part).

Inequalities

Inequalities in complex analysis describe regions or sets of points on the complex plane. They tell us where certain relationships hold true.
For the inequality \(2 < \operatorname{Re}(z-1) < 4\), we're looking at the real parts only.

• The inequality indicates a region on the complex plane between two vertical lines.
• The real part of the number is more than 2 but less than 4.

This creates an area where the solution set resides. Understanding inequalities helps us determine which part of the complex plane we should focus on.
The given inequality confines us to a specified strip, shaping the area where valid solutions to problems may be found.

Real Part of Complex Numbers

The real part of a complex number is just as important as the imaginary part. It's denoted as \(\operatorname{Re}(z)\), where \(z = x + yi\). Here, it simply means \(x\).
The real part lets us slice through the complex plane and analyze specific sections.For example, in \(z - 1\), if \(z = x + yi\), then \(z-1 = (x-1) + yi\), which tells us that\(\operatorname{Re}(z - 1) = x - 1\).
This vital function helps us more intuitively sketch solutions and understand a problem's constraints. Learning how to work with real parts widens our scope in tackling complex analysis challenges.

Domain in Complex Analysis

In complex analysis, a domain is a specific type of region on the complex plane. It's more than just a set of points, it needs to satisfy certain conditions.

• A domain is an open set, meaning it doesn't include its boundary points.
• It is also connected, meaning any two points within the domain can be joined by a continuous path lying entirely within the domain.

In our exercise, the set \(3 < x < 5\) and \(any\ y\) forms a domain. It's open because it doesn't include \(x=3\) or \(x=5\) (just the values in between), and it's connected because the strip is uninterrupted. Understanding domains is key to studying complex functions and their behaviors.