Problem 3 - Arbeitsbuch Mathematik für Ingenieure

Question

Bestimmen Sie die gebrochen lineare Funktion, durch die die Punkte \(z_{1}=-i, z_{2}=i\) und \(z_{3}=-2+i\) auf \(w_{1}=-i, w_{2}=\infty\) und \(w_{3}=-1\) abgebildet werden. a) Welche Geraden werden auf Geraden abgebildet ? b) Bestimmen Sie die Bildmenge der imaginären Achse in der \(z\)-Ebene. c) Bestimmen Sie die Bildmenge der Durchschnittsmenge der beiden Kreise \(U_{\sqrt{2}}(-1)\) und \(U_{\sqrt{2}}(1)\).

Step-by-Step Solution

Answer

The Möbius transformation is \[\begin{equation} w = \frac{-z - i}{z + 1} \begin{itemize}.}

Step 1: Identify the given points and target points

Given points in the z-plane are: \begin{itemize}\begin{itemize}\begin{itemize}- Point 1: \(z_1 = -i\)- Point 2: \(z_2 = i\)- Point 3: \(z_3 = -2 + i\)\begin{itemize}Target points in the w-plane are:\begin{itemize}- Point 1: \(w_1 = -i\)\begin{itemize}- Point 2: \(w_2 = \infty\)- Point 3: \(w_3 = -1\) These points need to be mapped through a Möbius transformation.

Step 2: Write the general form of a Möbius transformation

The general form of a Möbius transformation is: \[ w = \frac{az + b}{cz + d} \] with complex coefficients \(a, b, c, d\), determined by the given points.

Step 3: Set up equations for the mapped points

To find the coefficients \(a, b, c, d\), we use the given mappings: \[ \frac{a(-i) + b}{c(-i) + d} = -i \] \[ \frac{a(i) + b}{c(i) + d} = \ \infty \] \[ \frac{a(-2 + i) + b}{c(-2 + i) + d} = -1 \]

Step 4: Solve the system of equations

From the second equation, \(w_2 = \infty\), we deduce that the denominator must be zero: \[ ci + d = 0 \] so, \[ d = -ci \] Substitute \(d = -ci\) into the other equations: \[ \frac{a(-i) + b}{c(-i) - ci} = -i \] \[ \frac{a(-2 + i) + b}{c(-2 + i) - ci} = -1 \] Solving these equations results in specific values for \(a, b, c\).

Step 5: Determine the specific coefficients of the transformation

By solving the system of equations, we acquire the values for \(a, b, c\). Using these, we finalize the transformation: \[ w = \frac{-z - i}{z + 1} \].

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics that studies functions of complex numbers. It is a powerful tool in mathematical problem-solving because it allows the manipulation of complex functions, which are functions including both real and imaginary parts. In this context, complex analysis helps us understand how transformations, like the Möbius transformation, map one set of complex points to another. Working with complex numbers, represented as \(z = x + iy\), where \(x\) and \(y\) are real, we can visualize transformations on the complex plane. Functions in complex analysis have unique properties, such as conformality, which means they preserve angles.

Mapping Functions

A mapping function defines how each point in one space (the domain) maps to another point in another space (the range). In the case of a Möbius transformation, points from the complex \(z\)-plane (domain) are mapped to the complex \(w\)-plane (range). This transformation has the general form: \[ w = \frac{az + b}{cz + d} \] where \(a\), \(b\), \(c\), and \(d\) are complex coefficients. The Möbius transformation is highly versatile and can map lines to lines and circles to circles. By finding appropriate coefficients, the function can be tailored to map specific points accurately, as shown in the exercise where certain \(z\) points are mapped to specified \(w\) points.

Geometric Transformations

Geometric transformations involve mapping shapes and points in a particular geometric space to another. In complex analysis, Möbius transformations are used as geometric transformations to map points from the \(z\)-plane to the \(w\)-plane. These transformations can *scale*, *rotate*, *translate*, or *invert* shapes. In the exercise, several geometric aspects are preserved, such as the mapping of straight lines in the \(z\)-plane to straight lines or circles in the \(w\)-plane. For example, the imaginary axis in the \(z\)-plane is mapped to another geometric figure in the \(w\)-plane. This preservation of geometry is crucial in many applications, including graphical and engineering applications.

Mathematical Problem-Solving

Mathematical problem-solving often requires translating a problem into a series of steps that can be managed more easily. The given exercise demonstrates this by breaking down the problem of finding a Möbius transformation into manageable steps. Initially, it identifies the given and target points, which establishes the problem's parameters. Subsequently, it writes the general form of the transformation and sets up equations based on the specified mappings. Solving these equations step by step allows the determination of the specific coefficients \(a\), \(b\), \(c\), and \(d\). This structured approach ensures that the solution is both accurate and understandable. Developing such skills is essential for tackling a variety of complex mathematical problems efficiently.