Problem 11 - Complex Analysis

Question

(a) Let \(\mathbb{H}:=\\{z \in \mathbb{C} ; \quad \operatorname{Im} z>0\\}\) be the upper half-plane. Show: \(z \in \mathbb{H} \Longleftrightarrow-1 / z \in \mathbb{H}\). (b) Assume \(z, a \in \mathbb{C}\). Show: \(\quad|1-z \bar{a}|^{2}-|z-a|^{2}=\left(1-|z|^{2}\right)\left(1-|a|^{2}\right)\) Deduce: If \(|a|<1\), then $$ |z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1 $$

Step-by-Step Solution

Answer

(a) \( z \in \mathbb{H} \iff -1/z \in \mathbb{H} \). (b) The equivalences follow from the identities shown.

Step 1: Definition and Condition on Im(z)

Start with the upper half-plane definition: \( \mathbb{H} = \{ z \in \mathbb{C} \mid \operatorname{Im} z > 0 \} \).For any complex number \( z = x + yi \) where \( x, y \in \mathbb{R} \), \( z \in \mathbb{H} \) if \( y > 0 \).

Step 2: Analytic Expression for -1/z

Consider \( z \in \mathbb{H} \) and calculate \( -1/z \). Write \( z \) as \( x+yi \), then find the complex reciprocal: \[-\frac{1}{z} = -\frac{1}{x+yi} = -\frac{x - yi}{x^2 + y^2}\].

Step 3: Imaginary Part of -1/z

Compute \( \operatorname{Im}(-1/z) \) from step 2: \( \operatorname{Im}(-1/z) = \frac{-y}{x^2 + y^2} \). Since \( y > 0 \), \( \operatorname{Im}(-1/z) > 0 \). Thus, \( -1/z \in \mathbb{H} \).

Step 4: Showing the Equivalence

The steps above show that if \( z \in \mathbb{H} \), then \( -1/z \in \mathbb{H} \). Conversely, if \( -1/z \in \mathbb{H} \), similarly, \( \operatorname{Im}(z) > 0 \). Hence \( z \in \mathbb{H} \leftrightarrow -1/z \in \mathbb{H} \).

Step 5: Given Expression for Part (b)

To prove the given expression, recall:\(|1-z\bar{a}|^{2} - |z-a|^{2} = (1 - |z|^2)(1 - |a|^2)\).Start by expanding \(|1-z\bar{a}|^2\) and \(|z-a|^2\).

Step 6: Expanding the Squared Moduli

Compute each term:- \(|1-z\bar{a}|^2 = (1-z\bar{a})(1-\bar{z}a) = 1 - z\bar{a} - \bar{z}a + |z|^2|a|^2\).- \(|z-a|^2 = (z-a)(\bar{z}-\bar{a}) = |z|^2 - z\bar{a} - \bar{z}a + |a|^2\).

Step 7: Simplifying the Expression

Subtract the two expressions:- \(|1-z\bar{a}|^2 - |z-a|^2 = 1 - |z|^2|a|^2 - |z|^2 + |a|^2 = (1-|z|^2)(1-|a|^2)\).

Step 8: Conclude Conditions on |z|

Use the result from step 7 for equivalence statement:If \(|a| < 1\), then:-\( |z| < 1 \iff \left|\frac{z-a}{\bar{a}z -1}\right| < 1 \).-\( |z| = 1 \iff \left|\frac{z-a}{\bar{a}z -1}\right| = 1 \).

Step 9: Conclusion

Both parts have been shown: \( z \in \mathbb{H} \iff -1/z \in \mathbb{H} \).Also, the equivalences for \(|z|\) in terms of \(\left|\frac{z-a}{\bar{a}z -1}\right|\) are derived from the expressions proven.

Key Concepts

Upper Half-Plane

When exploring complex analysis, the upper half-plane is a fundamental concept to understand. It consists of all complex numbers whose imaginary part is greater than zero. For a complex number \( z \), written as \( x + yi \) (where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit), the upper half-plane \( \mathbb{H} \) is defined as:

• \( \mathbb{H} = \{ z \in \mathbb{C} \mid \operatorname{Im} z > 0 \} \)

This means if \( y > 0 \), then the complex number \( z \) is in the upper half-plane. In simpler terms, if a point lies above the real axis in the complex plane, it's part of this region. The significance of the upper half-plane in complex analysis arises due to its utility in various mathematical techniques, such as conformal mappings and the study of analytic functions.

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. This unique formulation allows them to represent a wider range of values than just real numbers alone. The standard form of a complex number is \( z = x + yi \), where \( x \) is the real part, and \( yi \) is the imaginary part. Here, \( i \) is the imaginary unit, defined as \( i^2 = -1 \). Complex numbers are essential for diverse fields of mathematics, including fractal geometry and complex dynamics. When analyzing complex numbers, we often work with their magnitude, or modulus, denoted as \(|z|\), which is calculated using the formula:

• \( |z| = \sqrt{x^2 + y^2} \)

Their rich structure allows solutions to equations that lack real solutions, enriching our mathematical toolset significantly.

Imaginary Part

One of the pivotal components of a complex number is its imaginary part. For any complex number \( z = x + yi \), the imaginary part is \( y \). This part dictates many properties of the number, such as dictating whether it lies within the upper half-plane. If \( y > 0 \), then the number is in the upper half-plane. The imaginary unit \( i \) is defined such that \( i^2 = -1 \), which gives rise to the unique characteristics of the imaginary component. By focusing on the imaginary part, we often discern key details about the behavior of complex functions in various analytic settings.

Analytic Expression

Developing an analytic expression for complex numbers involves formulating expressions that reflect their properties and relationships. In complex analysis, we often encounter expressions like \(-1/z\) for a complex number \(z\). This operation can drastically alter the properties of \(z\). For example, if \(z\) is in the upper half-plane, the operation \(-1/z\) can be shown to preserve this condition using an analytic approach. As seen in the exercise, calculating \(-1/z\) involves a few steps:

• Express \(z\) as \(x + yi\).
• Compute \(-1/z\) as: \[ -\frac{1}{z} = -\frac{x - yi}{x^2 + y^2} \]
• Find the imaginary part: \(\operatorname{Im}(-1/z) = \frac{-y}{x^2 + y^2}\).
• Since \(y > 0\), we see that \(\operatorname{Im}(-1/z) > 0\), confirming that \(-1/z\) is also in the upper half-plane.

This illustrates the subtleties involved in interpreting and analyzing complex analytic expressions and their implications in the broader context of mathematical theory.