Question

Die folgenden Teilmengen von \(C\) veranschauliche man sich in der komplexen Zahlenebene: a) Seien \(a, b \in \mathbb{C}, b \neq 0\), und $$ \begin{aligned} G_{0} &:=\\{z \in \mathrm{C} ;&\left.\operatorname{Im}\left(\frac{z-a}{b}\right)=0\right\\} \\ G_{+} &:=\\{z \in \mathrm{C} ;&\left.\operatorname{Im}\left(\frac{z-a}{b}\right)>0\right\\} \text { und } \\\ G_{-} &:=\\{z \in \mathcal{C} ;&\left.\operatorname{lm}\left(\frac{z-a}{b}\right)<0\right\\} \end{aligned} $$ b) Seien \(a, c \in \mathbb{R}\) und \(b \in \mathrm{C}\) mit \(b \bar{b}-a c>0\) $$ \boldsymbol{K}:=\\{z \in \mathbf{C} ; \quad a z \bar{z}+\bar{b} z+b \xi+c=0\\} $$ \(L:=\left\\{z \in \mathrm{C} ; \quad\left|z-\frac{\sqrt{2}}{2}\right|^{2} \cdot\left|z+\frac{\sqrt{2}}{2}\right|^{2}=\frac{1}{4}\right\\}\)

Short answer

The sets represent lines and half-planes in (a), and an ellipse and a specific locus in (b) in the complex plane.

Step-by-step solution

Understanding Condition A1

The set \(G_{0}\) is defined as \(\{z \in \mathbb{C} \; | \; \text{Im}\left(\frac{z-a}{b}\right)=0\}\). This implies that \(\frac{z-a}{b}\) is a real number. Thus, the set \(G_{0}\) represents all points \(z\) that lie on a line parallel to and offset from the direction of vector \(b\) in the complex plane.

Understanding Condition A2

The set \(G_{+}\) is defined as \(\{z \in \mathbb{C} \; | \; \text{Im}\left(\frac{z-a}{b}\right)>0\}\). This describes all points \(z\) for which the vector \(\frac{z-a}{b}\) has a positive imaginary component, indicating the points of the complex plane lying in the half-plane above the line described in Step 1.

Understanding Condition A3

The set \(G_{-}\) is defined as \(\{z \in \mathbb{C} \; | \; \text{Im}\left(\frac{z-a}{b}\right)<0\}\). This describes all points \(z\) for which the vector \(\frac{z-a}{b}\) has a negative imaginary component, indicating the points of the complex plane lying in the half-plane below the line described in Step 1.

Analyzing Condition B1

The equation \(a z \bar{z} + \bar{b} z + b \bar{z} + c = 0\) represents a conic section in the complex plane. Depending on the coefficients, this will typically represent either an ellipse, a hyperbola, or a parabola. The condition \(b \bar{b} - a c > 0\) ensures that the nature of this conic is an ellipse. Solve this quadratic equation to determine the exact nature of the curve in the complex plane.

Analyzing Condition B2

The set \(L\) is given by:\[L := \left\{z \in \mathbb{C} \; | \; \left| z - \frac{\sqrt{2}}{2} \right|^2 \cdot \left| z + \frac{\sqrt{2}}{2} \right|^2 = \frac{1}{4} \right\}.\]This describes a quartic equation where \(z\) lies on the set of points in the complex plane that maintains the product of these distances as \(\frac{1}{4}\). Simplifying this equation or visualizing may show symmetrical properties, potentially representing a more complex curve.

Key concepts

Complex Plane

The complex plane is a vital concept in complex analysis. Imagery and geometry are integrated into this two-dimensional plane, where each point represents a unique complex number. Unlike the traditional number line used for real numbers, the complex plane accommodates the imaginary unit. A complex number is expressed as \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit.

On this plane, the horizontal axis represents the real component, whereas the vertical axis portrays the imaginary component. This structure makes complex numbers visually intuitive, allowing mathematicians to solve equations graphically. For instance, in the original exercise, sets \( G_0 \), \( G_+ \), and \( G_- \) translate into geometric lines and regions on this plane, aiding in understanding their behavior and interactions.

Conic Sections

Conic sections are shapes formed by the intersection of a plane and a double cone. These curves, namely ellipses, parabolas, and hyperbolas, frequently appear in different mathematical contexts. In the realm of complex numbers, these sections can be formulated using complex equations.

For example, the equation like \( a z \bar{z} + \bar{b} z + b \bar{z} + c = 0 \) from our exercise is a complex representation of a conic section. The exercise also introduces the condition \( b \bar{b} - a c > 0 \), ensuring it forms an ellipse. Visualizing these complex conics on the plane expands the understanding of geometric properties and their transformations within complex analysis.

Imaginary Unit

The imaginary unit, denoted by \( i \), is the cornerstone of imaginary numbers and complex number calculations. It is defined such that \( i^2 = -1 \). This small definition forms the basis for expanding the real number system to include complex numbers.

In applications, the imaginary unit helps tag a "direction" to the imaginary part of a complex number, differentiating it from the real part. In visual terms, it aids in plotting numbers on the complex plane vertically, as opposed to merely linearly on the real number line. Understanding \( i \) and its properties boosts problem-solving capabilities in complex analysis and directly assists in analyzing the original exercise steps like those involving \( G_+ \) and \( G_- \).

Complex Numbers

Complex numbers are an extension of the real numbers, and they encompass both a real part and an imaginary part. Notated as \( z = x + yi \), where \( x \) and \( y \) are real, these numbers are a fundamental concept in complex analysis.

They extend our capability to solve equations, such as quadratic or polynomial equations, that do not have solutions strictly in the real numbers. Operations with complex numbers, like addition and multiplication, incorporate both real and imaginary components, adding a new dimension to mathematical problem-solving.

• **Addition & Subtraction:** Combine like components: \( (x_1 + y_1i) + (x_2 + y_2i) = (x_1 + x_2) + (y_1 + y_2)i \).
• **Multiplication:** Utilize distributive law, remembering \( i^2 = -1 \).

Through visualization on the complex plane, complex numbers help in understanding various properties, like those seen in sets \( G_0 \), \( G_+ \), and \( G_- \).