Problem 14
Question
Each \(z \in S^{1} \backslash\\{-1\\}\) $$ S^{1}:=\\{z \in \mathbb{C} ; \quad|z|=1\\} $$ can be uniquely represented in the form $$ z=\frac{1+\mathrm{i} \lambda}{1-\mathrm{i} \lambda}=\frac{1-\lambda^{2}}{1+\lambda^{2}}+\frac{2 \lambda}{1+\lambda^{2}} \mathrm{i} $$ with \(\lambda \in \mathbb{R}\).
Step-by-Step Solution
Verified Answer
Each \( z \) on \( S^1 \setminus \{-1\} \) can be uniquely represented in the two given forms with real \( \lambda \).
1Step 1: Define the Problem
We need to express each complex number \( z \) on the unit circle excluding \( -1 \) (i.e., \( z \in S^1 \setminus \{-1\} \)) uniquely in the form given by two different representations involving a real \( \lambda \).
2Step 2: Understand the Geometry of Unit Circle
The unit circle \( S^1 \) is defined as the set of all complex numbers \( z \) satisfying \(|z| = 1\). This means any \( z \) can be written as \( e^{i\theta} \), where \( \theta \) is the angle with the positive real axis.
3Step 3: Use Möbius Transformation
The given transformation is a Möbius transformation: \( z = \frac{1 + i\lambda}{1 - i\lambda} \). For this transformation, as \( \lambda \) varies over \( \mathbb{R} \), \( z \) traces the unit circle except the point \( -1 \).
4Step 4: Verify the Transformation
To ensure \( z \) is on the unit circle, we check \( |z| = 1 \). Calculate: \[|z| = \left|\frac{1 + i\lambda}{1 - i\lambda}\right| = \frac{|1 + i\lambda|}{|1 - i\lambda|}\] This evaluates to 1, reaffirming \( z \in S^1 \).
5Step 5: Simplify to Required Form
Simplify \( \frac{1 + i\lambda}{1 - i\lambda} \) into the second form using algebraic manipulations: - Multiply the numerator and the denominator by the conjugate of the denominator: \[z = \frac{(1 + i\lambda)(1 + i\lambda)^*}{(1 - i\lambda)(1 + i\lambda)^*}\]- This results in \[ z = \frac{(1 - \lambda^2) + 2\lambda i}{1 + \lambda^2}.\]
6Step 6: Verify the Final Expression
Compare our derived expression with the given form and see they match exactly: \[ z = \frac{1 - \lambda^2}{1 + \lambda^2} + \frac{2\lambda}{1 + \lambda^2}i.\]This shows the transformation was correctly applied.
7Step 7: Understanding Uniqueness of Representation
Given any \( z = e^{i\theta} eq -1 \), we evaluated that \( \theta = 2\arctan(\lambda) \). Since \( \theta \in (-\pi, \pi) \), \( \lambda \) is deterministically and uniquely determined by the tangent value.
Key Concepts
Unit CircleMöbius TransformationComplex Numbers
Unit Circle
The unit circle is a fundamental concept in complex analysis and trigonometry. It represents all the complex numbers that have an absolute value, or magnitude, of 1. When expressed mathematically, the unit circle is defined as:
The unit circle has many unique characteristics and is essential in understanding the behavior of complex numbers in various transformations. In this context, we focus on transformations that exclude the point \( -1 \) from the unit circle, highlighting the specific subset \( S^1 \setminus \{-1\} \), which means all points on the circle except for the point \( -1 \).
Unit circles serve as a foundation for many mathematical concepts.
It simplifies the behavior analysis of complex numbers when they undergo certain transformations like rotation and inversion.
By maintaining the property \( |z| = 1 \), it ensures consistent results regardless of the transformations applied.
- \( S^1 = \{ z \in \mathbb{C} : |z| = 1 \} \)
- \( z = e^{i\theta} \)
The unit circle has many unique characteristics and is essential in understanding the behavior of complex numbers in various transformations. In this context, we focus on transformations that exclude the point \( -1 \) from the unit circle, highlighting the specific subset \( S^1 \setminus \{-1\} \), which means all points on the circle except for the point \( -1 \).
Unit circles serve as a foundation for many mathematical concepts.
It simplifies the behavior analysis of complex numbers when they undergo certain transformations like rotation and inversion.
By maintaining the property \( |z| = 1 \), it ensures consistent results regardless of the transformations applied.
Möbius Transformation
A Möbius transformation is a function that maps a complex number plane into another complex number plane, often resulting in elegant, smooth changes.
The general form of a Möbius transformation is:
This transformation is particularly special because it can map circles to circles or straight lines. In the exercise, the transformation:
As \( \lambda \) sweeps over all real numbers, \( z \) traces out the entire unit circle excluding the point \( -1 \).
This method of exclusion is achieved due to the limitation \( \lambda eq 0 \), ensuring the transformation is defined over \( \mathbb{R} \).
Through this transformation, we convert the complex plane in a way that preserves angles and shapes, providing a useful tool for complex analysis.
The general form of a Möbius transformation is:
- \( z' = \frac{az + b}{cz + d} \)
This transformation is particularly special because it can map circles to circles or straight lines. In the exercise, the transformation:
- \( z = \frac{1 + i\lambda}{1 - i\lambda} \)
As \( \lambda \) sweeps over all real numbers, \( z \) traces out the entire unit circle excluding the point \( -1 \).
This method of exclusion is achieved due to the limitation \( \lambda eq 0 \), ensuring the transformation is defined over \( \mathbb{R} \).
Through this transformation, we convert the complex plane in a way that preserves angles and shapes, providing a useful tool for complex analysis.
Complex Numbers
Complex numbers are a core aspect of mathematics, defined as numbers of the form:
These numbers can be visually represented on the complex plane with \( a \) as the real part (along the x-axis) and \( b \) as the imaginary part (along the y-axis).
The magnitude or modulus is given by \(|z| = \sqrt{a^2 + b^2}\).
Complex numbers bring the benefit of handling two-dimensional quantities simultaneously, which is extremely useful in both geometry and physics.
They allow for operations that aren't possible with only real numbers, such as solving certain types of equations and performing specific transformations.
In our current exercise, we explore the beautiful transformation properties of complex numbers, particularly focusing on the uniqueness and representation of numbers on the unit circle.
We also delve deeper into how these numbers change when undergoing Möbius transformations, which can elegantly map the plane in structured, predictable ways.
- \( z = a + bi \)
These numbers can be visually represented on the complex plane with \( a \) as the real part (along the x-axis) and \( b \) as the imaginary part (along the y-axis).
The magnitude or modulus is given by \(|z| = \sqrt{a^2 + b^2}\).
Complex numbers bring the benefit of handling two-dimensional quantities simultaneously, which is extremely useful in both geometry and physics.
They allow for operations that aren't possible with only real numbers, such as solving certain types of equations and performing specific transformations.
In our current exercise, we explore the beautiful transformation properties of complex numbers, particularly focusing on the uniqueness and representation of numbers on the unit circle.
We also delve deeper into how these numbers change when undergoing Möbius transformations, which can elegantly map the plane in structured, predictable ways.
Other exercises in this chapter
Problem 13
Let \(n \geq 2\) be a natural number. There is no continuous function \(f: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}^{\bullet}\) with the two properties (a) \
View solution Problem 13
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Let \(n \geq 2\) be a natural number. There is no continuous function \(q_{n}: \mathbb{C} \rightarrow \mathbb{C}\) such that $$ \left(q_{n}(z)\right)^{n}=z \tex
View solution Problem 14
Let \(D=\\{z \in \mathbb{C} ; \quad-\pi
View solution