Problem 19 - Complex Analysis

Question

Let \(a, b\) and \(c\) be three different points in the RIEMANN sphere \(\bar{C}\). Show the existence of exactly one MöBius transformation \(M\) with the property: $$ M a=0, \quad M b=1, \quad M c=\infty $$ Hint. Consider $$ M z:=\frac{z-a}{z-c}: \frac{b-a}{b-c} $$ Note:. The expression on the right hand side of the above equation is the ubiquitous cross ratio of the four complex numbers \(z, a, b\) and \(c\), for short $$ \mathrm{CR}(z, a, b, c) \text { or } \quad[z, a, b, c] $$

Step-by-Step Solution

Answer

The Möbius transformation \(M(z) = \left( \frac{z-a}{z-c} \right) \Big/ \left( \frac{b-a}{b-c} \right)\) achieves the desired mappings.

Step 1: Understanding the Problem

We are given three distinct points on the Riemann sphere, denoted as \(a, b,\) and \(c\), and we must show that there is a Möbius transformation \(M\) for which the mappings are \(M(a) = 0\), \(M(b) = 1\), and \(M(c) = \infty\). We will use the cross ratio formula.

Step 2: Using Cross Ratio for Möbius Transformation

The hint provides the Möbius transformation \(M(z) = \left( \frac{z-a}{z-c} \right) \Big/ \left( \frac{b-a}{b-c} \right)\). This expression uses the cross ratio, denoted as \([z, a, b, c]\), which is invariant for Möbius transformations. Our task is to demonstrate that this transformation achieves the desired effect of mapping \(a, b, c\) to \(0, 1, \infty\).

Step 3: Mapping Verification

Evaluate the transformation at each point:1. When \(z = a\): \[ M(a) = \left( \frac{a-a}{a-c} \right) \Big/ \left( \frac{b-a}{b-c} \right) = 0 \] \(M(a) = 0\) as desired.2. When \(z = b\): \[ M(b) = \left( \frac{b-a}{b-c} \right) \Big/ \left( \frac{b-a}{b-c} \right) = 1 \] \(M(b) = 1\) as desired.3. When \(z = c\): \[ M(c) = \left( \frac{c-a}{c-c} \right) \Big/ \left( \frac{b-a}{b-c} \right) = \frac{\text{undefined}}{\left(\frac{b-a}{b-c}\right)} = \infty \] \(M(c) = \infty\) as desired.

Step 4: Uniqueness of the Transformation

The transformation \(M\) is uniquely determined by its properties of mapping three distinct points on the Riemann sphere to specific values \(0, 1, \infty\). This is a known property of Möbius transformations; mapping any three distinct points to another set of three distinct points uniquely defines a Möbius transformation.

Key Concepts

Riemann sphere

To fully grasp how Möbius transformations work, it's essential to understand the concept of the Riemann sphere. The Riemann sphere is a model of the extended complex plane, including all complex numbers and a point at infinity. This makes it an essential tool in complex analysis, especially when dealing with Möbius transformations.

• The Riemann sphere can be visualized as a globe, with the equator representing the complex plane.
• Its poles correspond to the point at infinity.
• This model helps simplify the analysis when points "go to infinity," a common occurrence in many complex transformation problems.

By using stereographic projection, one can map complex points onto this sphere seamlessly, allowing transformations that would be complex or undefined on a regular plane. This includes points such as infinity, which can be problematic. On the Riemann sphere, however, these points are just poles on the globe, making mathematical operations much more intuitive.

cross ratio

The cross ratio is a crucial concept in understanding Möbius transformations. It is an invariant under these transformations, meaning it remains unchanged, which is pivotal for solving problems involving transformations of three points.

• The cross ratio of four complex numbers \([z, a, b, c] = \frac{(z-a)(b-c)}{(z-c)(b-a)}\) expresses a special kind of symmetry important in complex analysis.
• Consider the transformation provided: \(M(z) = \frac{\left(\frac{z-a}{z-c}\right)}{\left(\frac{b-a}{b-c}\right)}\).

This expression represents a form of the cross ratio and ensures the transformation maps specific points as desired, like point \(a\) to 0, \(b\) to 1, and \(c\) to infinity. This number, sometimes referred to as the cross-ratio constant, ensures the transformation's consistency across points. This invariant property makes the cross ratio a powerful tool when determining characteristics and behaviors of complex transformations.

complex analysis

Complex analysis is the study of functions that map complex numbers to complex numbers. This analysis extends into the realm of transformations, like Möbius transformations, which play a critical role in many theoretical and practical applications.

• Möbius transformations can map an entire complex plane to another, maintaining specific properties like angle preservation and cross ratio invariance.
• These transformations are particularly powerful because they can be expressed as linear fractional transformations of the form \(M(z) = \frac{az + b}{cz + d}\), where \(ad - bc eq 0\).

In this exercise, complex analysis provides the foundational framework for understanding how and why Möbius transformations can uniquely map three distinct points on the Riemann sphere to 0, 1, and infinity. By employing properties such as invariance of the cross ratio, solutions become more intuitive and manageable within this field.