Problem 2 - Arbeitsbuch Mathematik für Ingenieure

Question

Auf welche Menge wird durch die gebrochen lineare Abbildung \(w=\frac{z}{z-1}\) der offene Kreisring zwischen den Kreisen \(K_{1}(0)\) und \(K_{2}(0)\) abgebildet? Hinweis: Beachten Sie, dass die reelle Achse auf sich abgebildet wird und dass die Abbildung winkeltreu ist.

Step-by-Step Solution

Answer

The open annulus between the circles maps onto itself or another similar annulus.

Step 1 - Understanding the transformation

The given transformation is defined by the function \( w = \frac{z}{z-1} \). It maps complex numbers z to another set of complex numbers w.

Step 2 - Analyzing the image of the circles

Consider the circles \(K_{1}(0)\) and \(K_{2}(0)\) with given radii. We need to find the image of these circles under the given transformation. First, note that the mapping is Möbius transformation, which is conformal (angle-preserving).

Step 3 - Image of the unit circle

For the unit circle \(\left|z\right|=1\), substitute \(z=e^{i\theta}\) into the transformation. This gives \(w=\frac{e^{i\theta}}{e^{i\theta}-1}\). Simplify this to find the image of every point on the unit circle.

Step 4 - Specific substitution and simplification

Let \(z=e^{i\theta} \). Then we get \(w = \frac{e^{i\theta}}{e^{i\theta} - 1} = \frac{1}{1 - e^{-i\theta}}\). Further simplify to see that \(w\) maps the unit circle to itself.

Step 5 - Real axis mapping

Since the transformation is \( w = \frac{z}{z-1} \), substitute \(z=x \) (x is real). This remains \( w = \frac{x}{x-1} \), which shows the real axis maps to itself due to the hint provided.

Step 6 - Mapping the annulus

Since the transformation maintains the real axis and maps the unit circle to itself, it indicates that any open annulus formed by \(0 < \text{radius of } K_{1} < \text{radius of } K_{2}\) will be mapped onto itself or another annulus.

Key Concepts

Complex Function Transformation

A Möbius transformation is a complex function of the form \( w = \frac{az + b}{cz + d} \). The transformation in our exercise \( w = \frac{z}{z-1} \) is a specific type of Möbius transformation because it can be considered with \( a = 1, b = 0, c = 1, d = -1 \). This type of transformation is fundamental in complex analysis because it maps points from the complex plane to other points following specific rules. Understanding these transformations involves analyzing how basic shapes like lines and circles move and change under the transformation. For example, we see that the unit circle (\|z\| = 1) is mapped into different regions of the complex plane.

When you use the given transformation, each point (z) in the plane is calculated and mapped to a new point (w). For example:

• A point (z = i) yields \(w = \frac{i}{i - 1}\).
• A point (z = -1) yields \(w = \frac{-1}{-2} = \frac{1}{2}\).

Understanding this transformation can help in grasping more complex function transformations which are used frequently in complex analysis.

Conformal Mapping

Möbius transformations are a subset of conformal mappings. Conformal mappings are special because they preserve angles. This means that if two curves intersect at a certain angle before the transformation, they will still intersect at the same angle after the transformation. This property is very powerful and useful in complex analysis when studying properties related to angles and shapes.

In our given transformation \(w = \frac{z}{z-1}\), we see that it preserves the angle at which the real axis intersects with the annulus. Consider, for example, how a right angle on the complex plane remains a right angle post-transformation.
Overall, conformal maps are essential for solving problems involving fluid dynamics, electromagnetism, and other areas of physics and engineering, ensuring that the structure and characteristics of functions are maintained while mapping.

Unit Circle Mapping

In the exercise, we analyzed the image of the unit circle under the Möbius transformation \(w = \frac{z}{z-1}\). To do this, first substitute \(z = e^{i\theta}\), where \(\|z\| = 1\) shows points on the unit circle. By doing this:
\[ w = \frac{e^{i\theta}}{e^{i\theta} - 1} \] Simplifying the equation, one gets:
\[ w = \frac{1}{1 - e^{-i\theta}} \]
This heated computation reveals interesting properties of the transformation.

Moreover, performing this transformation on the unit circle shows us how specific points move around the complex plane. The transformation maps each point on the circle onto another point on the circle:

• When \(z=e^{i0} = 1\), \(w = \frac{1}{1-1} = \text{undefined}\) (this corresponds to \(z=1\),
• When \(z=e^{i\frac{\theta}{2}} \) maps at specific corresponding ranges.

This method shows us without doubt how the unit circle maintains its structure but maps onto itself in a transformed way. This understanding forms the basis for further explorations in complex analysis transformations.