Problem 1 - Complex Analysis

Question

Which of the following subsets of \(\mathbb{C}\) are domains? (a) \(\left\\{z \in \mathbb{C} ; \quad\left|z^{2}-3\right|<1\right\\}\) (b) \(\left\\{z \in C ; \quad\left|z^{2}-1\right|<3\right\\}\) (c) \(\left\\{z \in \mathbb{C} ;\left.\quad|| z\right|^{2}-2 \mid<1\right\\}\) (d) \(\left\\{z \in C ; \quad\left|z^{2}-1\right|<1\right\\}\) (e) \(\\{z \in \mathbb{C} ; \quad z+|z| \neq 0\\}\) (f) \(\\{z \in \mathbb{C} ; 0

Step-by-Step Solution

Answer

Domains: (a), (b), (c), (d).

Step 1: Understand the Definition of a Domain

A domain in the context of complex numbers is an open, connected subset of \(\mathbb{C}\). Open means it does not include its boundary points and connected means there is a path between any two points in the set that lies entirely within the set.

Step 2: Analyze the Set in Part (a)

The set \(\{z \in \mathbb{C} ; |z^{2}-3|<1\}\) is open since it includes points within a fixed radius from the center \(z^2 = 3\). This region is connected because it forms an open disk in the complex plane. Thus, this subset is a domain.

Step 3: Analyze the Set in Part (b)

The set \(\{z \in C ; |z^{2}-1|<3\}\) is open for the same reasons as part (a). It is a disk centered about \(z^2 = 1\) with radius 3, which is connected. Therefore, it is a domain.

Step 4: Analyze the Set in Part (c)

The set \(\{z \in \mathbb{C} ; \left|| z|^{2}-2 \right|<1\}\) represents an open subset concentrating around the circle defined by \(|z|^2 = 2\), specifically between the two circles of radius \(\sqrt{1}\) and \(\sqrt{3}\). This forms an annulus, which is connected and open, making it a domain.

Step 5: Analyze the Set in Part (d)

The set \(\{z \in C ; |z^{2}-1|<1\}\) is also an open disk that is connected, making it a domain.

Step 6: Analyze the Set in Part (e)

The set \(\{z \in \mathbb{C} ; z+|z| eq 0\}\) contains all complex numbers, excluding the negative real numbers due to the condition involving their modulus. Although it is open, it is not connected because there is a gap, making it not a domain.

Step 7: Analyze the Set in Part (f)

The set \(\{z \in \mathbb{C} ; 0

Key Concepts

Domains in Complex Analysis

In complex analysis, a **domain** is a fundamental concept that refers to an open and connected subset of the complex plane, denoted as \(\mathbb{C}\). The idea of a domain helps us understand where complex functions are defined and how their behavior is analyzed across the complex plane.
- **Open:** A set is considered open if it does not include its boundaries. This means you can move slightly in any direction within the set without leaving it.
- **Connected:** A set is connected if you can draw a path between any two points in the set that remains entirely within it.
Understanding if a subset of \(\mathbb{C}\) is a domain is crucial since many theorems in complex analysis, such as the Cauchy-Goursat theorem and Morera's theorem, require the function to be defined on a domain.

Open and Connected Sets

To classify a subset of the complex plane as a domain, it must be both **open** and **connected**. Let’s explore these terms in the context of complex analysis.

• Open Sets: This implies there are no boundary points included in the set. For example, the set \(\{ z \in \mathbb{C} : |z| < 1 \}\) is open because it's describing all points within, but not on, the circle at some radius from a point.
• Connected Sets: Being connected means you can move from one point to another in the set without "jumping" over any missing points. It creates a seamless path, indicating no gaps or separations.

For a set to be considered a domain in complex analysis, it must satisfy both of these conditions, ensuring that complex functions defined over these sets behave predictably and are analyzable.

Complex Plane Geometry

The complex plane, also known as the **Argand plane**, is a two-dimensional plane used to represent complex numbers. Each complex number \(z = x + yi\) is associated with a point \((x, y)\) in this complex plane.

• Real Axis: It represents the real part of complex numbers, running horizontally across the plane.
• Imaginary Axis: It represents the imaginary part, running vertically.

Using the complex plane geometry, we visualize complex numbers as points or vectors, making it easier to grasp operations like addition, subtraction, and multiplication.
Understanding these geometrical interpretations aids in conceptualizing more complex ideas such as the modulus and argument of a complex number.

Open Disks

An **open disk** in the complex plane is a crucial concept indicating a round region without its boundary circle.
- It's defined as \(\{ z \in \mathbb{C} : |z - a| < r \}\), where \(a\) is the center and \(r\) is the radius.
Open disks are important because they are simple examples of open, connected sets—making them qualify as domains in complex analysis. They help illustrate basic concepts and are foundational for more complex constructs.
Notice how open disks embody both openness and connectedness. All points within such a disk are reachable without encountering boundary constraints, satisfying the definition needed for analyzing complex functions.

Annulus in Complex Plane

An **annulus** is a ring-shaped region in the complex plane defined by two concentric circles. Given by \(\{ z \in \mathbb{C} : r_1 < |z - a| < r_2 \}\), where \(r_1\) and \(r_2\) are the radii of the inner and outer circle, respectively.
An annulus is:
- **Open:** Because we exclude the boundaries defined by both circles.
- **Connected:** Since any point within it can be reached by continuous paths that do not cross into or out of the circle's boundary.
This setup makes annuli perfect examples of domains in complex analysis where operations can be performed smoothly. They help in advancing the understanding of concepts like contour integration and the function behavior in different `layers` of the complex plane.

Next

Problem 2

Other exercises in this chapter

Problem 2

Let \(\alpha:[0, \pi] \rightarrow \mathbb{C}\) be defined by $$ \alpha(t):=\exp (\mathrm{i} t) $$ and \(\beta:[0,2] \rightarrow \mathbb{C}\) by $$ \beta(t)= \be

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Problem 2

Let \(z_{0}, \ldots, z_{N} \in \mathbb{C}(N \in \mathbb{N}) .\) Define the line segments connecting \(z_{\nu}\) with \(z_{2+1}\) \((\nu=0,1, \ldots, N-1) \mathr

View solution

Problem 4

Sketch the following curve \(\alpha\) ("figure eight") $$ \alpha(t):=\left\\{\begin{aligned} 1-\exp (\text { it }) & \text { for } t \in[0,2 \pi] \\ -1+\exp (-\

View solution

Key Concepts

Domains in Complex Analysis

In complex analysis, a **domain** is a fundamental concept that refers to an open and connected subset of the complex plane, denoted as \(\mathbb{C}\). The idea of a domain helps us understand where complex functions are defined and how their behavior is analyzed across the complex plane.
- **Open:** A set is considered open if it does not include its boundaries. This means you can move slightly in any direction within the set without leaving it.
- **Connected:** A set is connected if you can draw a path between any two points in the set that remains entirely within it.
Understanding if a subset of \(\mathbb{C}\) is a domain is crucial since many theorems in complex analysis, such as the Cauchy-Goursat theorem and Morera's theorem, require the function to be defined on a domain.

Open and Connected Sets

To classify a subset of the complex plane as a domain, it must be both **open** and **connected**. Let’s explore these terms in the context of complex analysis.

• Open Sets: This implies there are no boundary points included in the set. For example, the set \(\{ z \in \mathbb{C} : |z| < 1 \}\) is open because it's describing all points within, but not on, the circle at some radius from a point.
• Connected Sets: Being connected means you can move from one point to another in the set without "jumping" over any missing points. It creates a seamless path, indicating no gaps or separations.

For a set to be considered a domain in complex analysis, it must satisfy both of these conditions, ensuring that complex functions defined over these sets behave predictably and are analyzable.

Complex Plane Geometry

The complex plane, also known as the **Argand plane**, is a two-dimensional plane used to represent complex numbers. Each complex number \(z = x + yi\) is associated with a point \((x, y)\) in this complex plane.

• Real Axis: It represents the real part of complex numbers, running horizontally across the plane.
• Imaginary Axis: It represents the imaginary part, running vertically.

Using the complex plane geometry, we visualize complex numbers as points or vectors, making it easier to grasp operations like addition, subtraction, and multiplication.
Understanding these geometrical interpretations aids in conceptualizing more complex ideas such as the modulus and argument of a complex number.

Open Disks

An **open disk** in the complex plane is a crucial concept indicating a round region without its boundary circle.
- It's defined as \(\{ z \in \mathbb{C} : |z - a| < r \}\), where \(a\) is the center and \(r\) is the radius.
Open disks are important because they are simple examples of open, connected sets—making them qualify as domains in complex analysis. They help illustrate basic concepts and are foundational for more complex constructs.
Notice how open disks embody both openness and connectedness. All points within such a disk are reachable without encountering boundary constraints, satisfying the definition needed for analyzing complex functions.

Annulus in Complex Plane

An **annulus** is a ring-shaped region in the complex plane defined by two concentric circles. Given by \(\{ z \in \mathbb{C} : r_1 < |z - a| < r_2 \}\), where \(r_1\) and \(r_2\) are the radii of the inner and outer circle, respectively.
An annulus is:
- **Open:** Because we exclude the boundaries defined by both circles.
- **Connected:** Since any point within it can be reached by continuous paths that do not cross into or out of the circle's boundary.
This setup makes annuli perfect examples of domains in complex analysis where operations can be performed smoothly. They help in advancing the understanding of concepts like contour integration and the function behavior in different `layers` of the complex plane.