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Magazine Chapter 2: Problem 1
Welche der folgenden 'Teilmengen von \(\mathrm{C}\) sind Gebiete?
a) \(\left\\{z \in \mathrm{C} ; \quad\left|z^{2}-3\right|<1\right\\}\),
b) \(\left\\{z \in \mathbb{C}: \quad z^{2}-1 \mid<3\right\\}\),
c) \(\left\\{z \in \mathrm{C} ; \quad|z|^{2}-2 \mid<1\right\\}\),
d) \(\left\\{z \in \mathrm{C} ; \quad\left|z^{2}-1\right|<1\right\\}\)
e) \(\\{z \in \mathbb{C} ; \quad z+|z| \neq 0\\}\),
f) \(\\{z \in \mathrm{C} ; 0
Short Answer
Expert verified
The subsets (a), (b), (c), and (d) are Gebiete.
Step by step solution
01
Understanding What a 'Gebiet' Means
A 'Gebiet' in the complex plane is a domain, which generally is an open and connected subset of the complex numbers. The open condition means for every point in the domain, there exists an epsilon-neighborhood within the domain. The connected condition ensures there is a path within the subset between any two points in it.
02
Analyze Option (a)
The set described by option (a) is \( \{z \in \mathbb{C} ; \ |z^{2}-3|<1 \}\). This inequality describes an open circle in the complex plane with center at \( \sqrt{3} \) and radius 1 (due to the square root principle and symmetry, center can also be at \(-\sqrt{3}\)). This is both open and connected, making it a domain or 'Gebiet.'
03
Analyze Option (b)
The set defined by option (b) is \( \{z \in \mathbb{C} : |z^{2}-1|<3 \} \). This represents an open set around the points where \( z^{2} = 1 \), including a family of points satisfying the inequality, forming an open and connected region. Hence, it is a 'Gebiet.'
04
Analyze Option (c)
The set in option (c) is \( \{z \in \mathbb{C} ; \ ||z|^{2}-2|<1 \} \). Simplifying the inequality \(|z|^{2}-2 < 1\), we get \(|z|^{2} < 3\). This describes the inside of a circle of radius \(\sqrt{3}\)", centered at the origin, excluding the boundary, making it an open and connected domain 鈥� a 'Gebiet.'
05
Analyze Option (d)
The expression \( \{z \in \mathbb{C} ; |z^{2}-1|<1 \} \) in option (d) defines an open set with a center at \(1\) and radius less than \(1\). It forms an open circle, hence it is also open and connected, making it a 'Gebiet.'
06
Analyze Option (e)
The set \( \{z \in \mathbb{C}; z+|z| eq 0 \} \) is open because it omits the curve \( z + |z| = 0 \), but it is not necessarily connected as it excludes a strip where this equality holds. Hence, it is not a 'Gebiet.'
07
Analyze Option (f)
The description in option (f) involves an open square-sector \(\{0 < x < 1, 0 < y < 1\} - \bigcup_{n=2}^{\infty} \{x+iy; x=1/n, 0 < y \leq 1/2\}\). While it removes countably many vertical lines, the remainder is still open in the complex plane. However, the removal of these lines cuts through horizontally from \(y=0.5\), making it disconnected. Consequently, it is not a 'Gebiet.'
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Numbers
Complex numbers are an extension of the real numbers, introducing the concept of the imaginary unit. The imaginary unit, denoted as \(i\), satisfies the equation \(i^2 = -1\). A complex number is typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. If \(b = 0\), the complex number is purely real. Conversely, if \(a = 0\), the number is purely imaginary.
- The real part of a complex number \(a + bi\) is \(a\).
- The imaginary part is \(b\).
Open Sets
An open set in the context of complex analysis is a vital concept. To say a set is open means intuitively that you can move a bit in any direction from any point in the set, and you'll still be inside the set. More formally, a set \(S\) in the complex plane is open if, for every point \(z\) in \(S\), there's a small "neighborhood" around \(z\) that is entirely contained within \(S\). This neighborhood can be envisioned as a circle around \(z\) with a radius \(\epsilon > 0\), where every point in this circle also belongs to \(S\).
Open sets are an essential building block in topology, allowing for shapes without defined boundaries, much like a circle without its outer edge.
Open sets are an essential building block in topology, allowing for shapes without defined boundaries, much like a circle without its outer edge.
Connected Sets
In complex analysis, a connected set means that for any two points within the set, there exists a continuous path within the set that connects them. Connected sets are important because they ensure that the set is "in one piece," without any disjoint parts.
- A set is connected if there is no separation into disjoint open subsets.
- This characteristic ensures there are paths between any two points in the set.
Complex Plane
The complex plane is a mathematical concept that extends the idea of the Cartesian coordinate plane to include complex numbers. Every complex number \(z = a + bi\) corresponds to a point in the complex plane, with \(a\) as the x-coordinate (real part) and \(b\) as the y-coordinate (imaginary part). The plane is often described using the terms "real axis" for the horizontal line and "imaginary axis" for the vertical line.
- The origin of the complex plane is the point \((0, 0)\), representing the complex number zero.
- The distance of any point from the origin is called the modulus of the complex number, calculated as \(|z| = \sqrt{a^2 + b^2}\).
