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Chapter 1: Problem 11

(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 $$ |z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1 $$

Short Answer

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

Step by step solution

01

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 \( y > 0 \).
02

Analytic Expression for -1/z

Consider \( z \in \mathbb{H} \) and calculate \( -1/z \). Write \( z \) as \( x+yi \), then find the complex reciprocal: \[-\frac{1}{z} = -\frac{1}{x+yi} = -\frac{x - yi}{x^2 + y^2}\].
03

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, \( -1/z \in \mathbb{H} \).
04

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 \( z \in \mathbb{H} \leftrightarrow -1/z \in \mathbb{H} \).
05

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 \(|z-a|^2\).
06

Expanding the Squared Moduli

Compute each term:- \(|1-z\bar{a}|^2 = (1-z\bar{a})(1-\bar{z}a) = 1 - z\bar{a} - \bar{z}a + |z|^2|a|^2\).- \(|z-a|^2 = (z-a)(\bar{z}-\bar{a}) = |z|^2 - z\bar{a} - \bar{z}a + |a|^2\).
07

Simplifying the Expression

Subtract the two expressions:- \(|1-z\bar{a}|^2 - |z-a|^2 = 1 - |z|^2|a|^2 - |z|^2 + |a|^2 = (1-|z|^2)(1-|a|^2)\).
08

Conclude Conditions on |z|

Use the result from step 7 for equivalence statement:If \(|a| < 1\), then:-\( |z| < 1 \iff \left|\frac{z-a}{\bar{a}z -1}\right| < 1 \).-\( |z| = 1 \iff \left|\frac{z-a}{\bar{a}z -1}\right| = 1 \).
09

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For all \(z \in \mathbb{C}\) $$ \lim _{n \rightarrow \infty}(1+z / n)^{n}=\exp (z) $$ More generally: For each sequence \(\left(z_{n}\right), z_{n} \in \mathbb{C}\) with \(\lim _{n \rightarrow \infty} z_{n}=z\) we have $$ \lim _{n \rightarrow \infty}\left(1+z_{n} / n\right)^{n}=\exp (z) $$

Let \(D=\\{z \in \mathbb{C} ; \quad-\pi<\operatorname{Im} z<\pi, 0<\operatorname{Re} z

Determine all the harmonic functions $$ u: \mathbb{C}^{\bullet}=\mathbb{R}^{2} \backslash\\{(0,0)\\} \longrightarrow \mathbb{R} $$ that depend only on \(r:=\sqrt{x^{2}+y^{2}} .\)

If the function \(f: \mathbb{C} \rightarrow \mathbb{C}\) is complex differentiable at all points \(z \in \mathbb{C}\) and takes on only real or pure imaginary values, then \(f\) is constant.

For \(z=x+\mathrm{i} y, w=u+\mathrm{i} v\), with \(x, y, u, v \in \mathbb{R}\), the standard scalar product in the \(\mathbb{R}\)-vector space \(\mathbb{C}=\mathbb{R} \times \mathbb{R}\) with respect to the basis \((1, \mathrm{i})\) is defined by $$ \langle z, w\rangle:=\operatorname{Re}(z \bar{w})=x u+y v $$ Verify by direct calculation that, for \(z, w \in \mathbb{C}\) $$ \langle z, w\rangle^{2}+\langle\mathrm{i} z, w\rangle^{2}=|z|^{2}|w|^{2} $$ and infer from this the CAUCHY-SCHWARZ Inequality in \(\mathbb{R}^{2}\) : $$ |\langle z, w\rangle|^{2}=|x u+y v|^{2} \leq|z|^{2}|w|^{2}=\left(x^{2}+y^{2}\right)\left(u^{2}+v^{2}\right) $$ In addition, show the following identities for \(z, w \in \mathbb{C}\) by direct calculation: In addition, show the following identities for \(z, w \in \mathbb{C}\) by direct calculation: \(\begin{aligned}|z+w|^{2} &=|z|^{2}+2\langle z, w\rangle+|w|^{2} & &(\text { cosine law }) \\\|z-w|^{2} &=|z|^{2}-2\langle z, w\rangle+|w|^{2}, & & \\\|z+w|^{2}+|z-w|^{2} &=2\left(|z|^{2}+|w|^{2}\right) & & \text { (parallelogram law) } \end{aligned}\) (parallelogram law). Further, show that for each pair \((z, w) \in \mathbb{C}^{\bullet} \times \mathbb{C}^{\bullet}\) there is a unique real number \(\omega:=\omega(z, w) \in]-\pi, \pi]\) with $$ \cos \omega=\cos \omega(z, w)=\frac{\langle z, w\rangle}{|z||w|} $$ and $$ \sin \omega=\sin \omega(z, w)=\frac{\langle\mathrm{i} z, w\rangle}{|z||w|} $$ \(\omega=\omega(z, w)\) is called the oriented angle between \(z\) and \(w\) and will often be denoted by \(\angle(z, w)\) Show: \(\quad \angle(1, \mathrm{i})=\pi / 2, \angle(\mathrm{i}, 1)=-\pi / 2=-\angle(1, \mathrm{i}) .\)
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