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) \(\left\\{z \in \mathbb{C} ; \quad\left|z^{2}-1\right|<3\right\\}\), (c) \(\left\\{z \in \mathbb{C} ;\left.\quad|| z\right|^{2}-2 \mid<1\right\\}\), (d) \(\left\\{z \in \mathbb{C} ; \quad\left|z^{2}-1\right|<1\right\\}\) (e) \(\\{z \in \mathbb{C} ; \quad z+|z| \neq 0\\}\) (f) \(\\{z \in \mathbb{C} ; 0

Step-by-Step Solution

Answer

Subsets (a), (b), (c), (d), and (e) are domains; (f) is not a domain.

Step 1: Understand the Definition of a Domain

In complex analysis, a domain is a non-empty open connected subset of \(\mathbb{C}\). This means the set must be open, without any boundary points, and every point within the set must be reachable from any other point in the set without leaving it.

Step 2: Analyze Subset (a)

The subset (a) \(\left\{z \in \mathbb{C} ; \quad\left|z^{2}-3\right|<1\right\}\) represents the interior of a circle centered at 3, with radius 1 in the complex plane. Since this set is open and connected, it qualifies as a domain.

Step 3: Analyze Subset (b)

The subset (b) \(\left\{z \in \mathbb{C} ; \quad\left|z^{2}-1\right|<3\right\}\) is also the interior of a circle, centered at 1 with radius 3. This set is open and connected, meaning it is a domain.

Step 4: Analyze Subset (c)

The subset (c) \(\left\{z \in \mathbb{C} ;\left.\quad|| z\right|^{2}-2 \mid<1\right\}\) describes points whose square modulus lies between 1 and 3, which forms an annulus (an open ring). This set is open and connected, qualifying as a domain.

Step 5: Analyze Subset (d)

The subset (d) \(\left\{z \in \mathbb{C} ; \quad\left|z^{2}-1\right|<1\right\}\) is again a circular region in the complex plane, a smaller circle centered at 1 with radius 1. This is an open and connected region, so it is also a domain.

Step 6: Analyze Subset (e)

The subset (e) \(\{z \in \mathbb{C} ; \quad z+|z| eq 0\}\) includes all points where the sum of \(z\) and its modulus is not zero. This condition creates an open set in the complex plane that excludes the negative real axis; the set is open and connected (since it retains path-connectedness without the negative real axis), making it a domain.

Step 7: Analyze Subset (f)

The subset (f) "\(\{z \in \mathbb{C} ; 0

Key Concepts

Open Sets

In complex analysis, an open set is a crucial concept that forms part of the definition of a domain. An open set in the complex plane is a set wherein each point within the set has a neighborhood that is entirely contained in the set itself. This means that there are no boundary points in an open set; every point must be an inner point.

To visualize this, imagine a circle on the complex plane. If the circle includes its boundary, it's not an open set. But if it only consists of the interior points, excluding the edge, then it is open. Such a set allows for movement throughout without stepping outside its boundaries. In mathematical terms, a set \(U\) is open if, for every point \(z\) in \(U\), there exists a radius \(r > 0\) such that all points within distance \(r\) from \(z\) are also in \(U\).

Understanding open sets helps us identify domains in complex analysis, which are non-empty open connected subsets of the complex numbers \(\mathbb{C}\).

Connected Sets

Connected sets in the complex plane are sets where any two points can be joined by a path that remains entirely within the set. This means there are no isolated parts; everything is reachable from everywhere else within the set.

Imagine holding a piece of string at two points in a field; if you can lay it fully without leaving the area, then the field is connected. Mathematically, a set \(U\) in the complex plane is connected if, for any points \(z_1, z_2 \in U\), you can find a continuous path within \(U\) from \(z_1\) to \(z_2\).

Connectedness is significant because a domain must be both open and connected. If you remove parts of a set, like cutting across our imaginary field, you could disrupt connectedness, just as with subset (f) in the exercise, where removed segments broke the open path, causing it not to qualify as a domain.

Complex Plane

The complex plane is an essential concept when working with complex numbers. It's a two-dimensional plane where each point represents a complex number. The horizontal axis, known as the real axis, portrays the real part of complex numbers, while the vertical axis, the imaginary axis, represents the imaginary part.

Think of the complex plane as a mapping tool, allowing us to visualize and work with complex numbers just like coordinates in geometry. A complex number \(z\) is expressed as \(x + yi\), where \(x\) is the real part and \(yi\) is the imaginary part. On the complex plane, \(x\) determines how far along the real axis the number lies, and \(y\) determines the position along the imaginary axis.

Working within the complex plane lets us explore more intricate structures, like those discussed in the exercise. Each subset of \(\mathbb{C}\) in the problems represents a region in this plane, examining how open or connected they are to decide if they form domains.

Complex Numbers

Complex numbers are numbers that extend the idea of the one-dimensional number line into the two-dimensional complex plane by including imaginary numbers. A complex number is written in the form \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) represents the imaginary unit with the property that \(i^2 = -1\).

Complex numbers have wide applications, from electrical engineering to fluid dynamics, because they simplify the handling of two-dimensional problems and oscillations.

• The real part \(x\) and imaginary part \(y\) can be thought of as coordinates in the complex plane.
• They allow operations like addition, subtraction, and multiplication, analogously to vector operations.
• The modulus of a complex number, \(|z| = \sqrt{x^2 + y^2}\), shows its distance from the origin in the complex plane.

Understanding these aspects of complex numbers is vital for grasping more complex concepts like the one in the exercise, where complex numbers define and describe domains in their subsets.

Next

Problem 2

Other exercises in this chapter

Problem 2

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

View solution

Problem 2

Let \(z_{0}, \ldots, z_{N} \in \mathbb{C}(N \in \mathbb{N}) .\) Define the segments connecting \(z_{\nu}\) to \(z_{\nu+1},(\nu=\) \(0,1, \ldots, N-1)\), by $$ \

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

Open Sets

In complex analysis, an open set is a crucial concept that forms part of the definition of a domain. An open set in the complex plane is a set wherein each point within the set has a neighborhood that is entirely contained in the set itself. This means that there are no boundary points in an open set; every point must be an inner point.

To visualize this, imagine a circle on the complex plane. If the circle includes its boundary, it's not an open set. But if it only consists of the interior points, excluding the edge, then it is open. Such a set allows for movement throughout without stepping outside its boundaries. In mathematical terms, a set \(U\) is open if, for every point \(z\) in \(U\), there exists a radius \(r > 0\) such that all points within distance \(r\) from \(z\) are also in \(U\).

Understanding open sets helps us identify domains in complex analysis, which are non-empty open connected subsets of the complex numbers \(\mathbb{C}\).

Connected Sets

Connected sets in the complex plane are sets where any two points can be joined by a path that remains entirely within the set. This means there are no isolated parts; everything is reachable from everywhere else within the set.

Imagine holding a piece of string at two points in a field; if you can lay it fully without leaving the area, then the field is connected. Mathematically, a set \(U\) in the complex plane is connected if, for any points \(z_1, z_2 \in U\), you can find a continuous path within \(U\) from \(z_1\) to \(z_2\).

Connectedness is significant because a domain must be both open and connected. If you remove parts of a set, like cutting across our imaginary field, you could disrupt connectedness, just as with subset (f) in the exercise, where removed segments broke the open path, causing it not to qualify as a domain.

Complex Plane

The complex plane is an essential concept when working with complex numbers. It's a two-dimensional plane where each point represents a complex number. The horizontal axis, known as the real axis, portrays the real part of complex numbers, while the vertical axis, the imaginary axis, represents the imaginary part.

Think of the complex plane as a mapping tool, allowing us to visualize and work with complex numbers just like coordinates in geometry. A complex number \(z\) is expressed as \(x + yi\), where \(x\) is the real part and \(yi\) is the imaginary part. On the complex plane, \(x\) determines how far along the real axis the number lies, and \(y\) determines the position along the imaginary axis.

Working within the complex plane lets us explore more intricate structures, like those discussed in the exercise. Each subset of \(\mathbb{C}\) in the problems represents a region in this plane, examining how open or connected they are to decide if they form domains.

Complex Numbers

Complex numbers are numbers that extend the idea of the one-dimensional number line into the two-dimensional complex plane by including imaginary numbers. A complex number is written in the form \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) represents the imaginary unit with the property that \(i^2 = -1\).

Complex numbers have wide applications, from electrical engineering to fluid dynamics, because they simplify the handling of two-dimensional problems and oscillations.

• The real part \(x\) and imaginary part \(y\) can be thought of as coordinates in the complex plane.
• They allow operations like addition, subtraction, and multiplication, analogously to vector operations.
• The modulus of a complex number, \(|z| = \sqrt{x^2 + y^2}\), shows its distance from the origin in the complex plane.

Understanding these aspects of complex numbers is vital for grasping more complex concepts like the one in the exercise, where complex numbers define and describe domains in their subsets.