Problem 7 - Arbeitsbuch Mathematik für Ingenieure

Question

Geben Sie alle auf C regulären Funktionen \(f\) an, die \(|f(z)|=1\) für \(z \in \mathbb{C}\) erfüllen.

Step-by-Step Solution

Answer

All such functions are of the form \(f(z) = e^{i\theta}\), where \(\theta\) is a real constant.

Step 1 - Understand the Problem

We need to find all functions \(f\) that are regular (holomorphic) on the complex plane \(\mathbb{C}\) and satisfy \(|f(z)| = 1\) for all \(z \in \mathbb{C}\).

Step 2 - Consider the Condition \(|f(z)| = 1\)

The condition \(|f(z)| = 1\) implies that the function \(f(z)\) must lie on the unit circle in the complex plane for all \(z\) in \(\mathbb{C}\). That means \(f(z)\) can be expressed as \(e^{i\theta(z)}\) where \(\theta(z)\) is a real-valued function.

Step 3 - Analyze Regular Functions

For \(f(z) = e^{i\theta(z)}\) to be holomorphic on \(\mathbb{C}\), \(\theta(z)\) must also be a holomorphic function. However, if \(\theta(z)\) were non-constant and holomorphic, it would map \(\mathbb{C}\) onto a dense subset in \(\mathbb{C}\), which contradicts the fact that \(\theta(z)\) is real-valued. Therefore, \(\theta(z)\) must be a constant function.

Step 4 - Determine the Constant Function

Since \(\theta(z) = \theta\) is constant, the regular function \(f(z) = e^{i\theta}\) is also constant for all \(z\). Thus, the only holomorphic functions on \(\mathbb{C}\) that satisfy \(|f(z)| = 1\) must be of the form \(f(z) = e^{i\theta}\), where \(\theta\) is a constant real number.

Key Concepts

complex plane

In complex analysis, the complex plane is a fundamental concept, providing a way to visualize and work with complex numbers. Imagine a plane where each point corresponds to a complex number, represented as a pair \((x, y)\) where \(x\) and \(y\) are real numbers. The complex number is then written as \(z = x + yi\).

Key aspects of the complex plane include:

• The horizontal axis (real axis) represents the real part of the complex number.
• The vertical axis (imaginary axis) represents the imaginary part.
• Complex numbers can be added, subtracted, multiplied, and divided on this plane.

The complex plane lets us apply geometric intuition to complex numbers, simplifying the understanding of their properties and relationships.

unit circle

The unit circle is a crucial concept within the complex plane. It's the set of all points in the complex plane that are exactly one unit away from the origin (0,0). Mathematically, the unit circle can be defined by the equation \(|z| = 1\), where \(z\) is a complex number.

Important properties include:

• All complex numbers on the unit circle have a magnitude (or modulus) of 1.
• Any complex number \(z\) on the unit circle can be expressed in the form \((z = e^{i\theta})\), where \(\theta\) is a real-valued angle measured in radians.

This means that for a function to lie on the unit circle, it must fit this exponential form, leading to significant implications for regular (holomorphic) functions.

constant function analysis

Constant function analysis is particularly relevant when examining holomorphic functions that lie on the unit circle. A function \(f(z)\) is constant if its output does not change regardless of the input \(z\).

Key points include:

• For \(f(z) = e^{i\theta}\) to be holomorphic over the entire complex plane, \(\theta\) cannot depend on \(z\) and must be constant.
• If \(\theta(z)\) were non-constant and holomorphic, it would map the complex plane to a dense subset, contradicting the real-valued constraint.

Thus, \(\theta\) must remain constant, resulting in \(f(z) = e^{i\theta}\) being the only suitable form for a holomorphic function that stays on the unit circle.

real-valued functions

Real-valued functions assign a real number to each input from their domain. In the context of holomorphic functions on the unit circle, it's crucial to understand that any variation depending on the input \(z\) would impact the function's holomorphic nature.

Key details include:

• In the given problem, \(\theta\) must be real-valued to ensure \(f(z) = e^{i\theta(z)}\) stays on the unit circle.
• A real-valued, holomorphic function that's non-constant would be impossible over the full complex plane without creating contradictions.

Consequently, \(\theta\) must be constant, reinforcing that \(f(z)\) is constant, of the form \(f(z) = e^{i\theta}\). This approach ensures the function remains real-valued while respecting holomorphy.