Question

a) Sei Hl \(:=\\{z \in \mathrm{C} ; \quad \operatorname{lm} z>0\\}\) die obere Halbebene. Man zeige: \(z \in \mathrm{H} \Leftrightarrow-1 / z \in \mathrm{H}\) b) Seien \(z, a \in \mathbb{C}\) Man zeige: \(\quad|1-z \bar{a}|^{2}-|z-a|^{2}=\left(1-|z|^{2}\right)\left(1-|a|^{2}\right) .\) Mon folgere: Ist \(|a|<1\), dann gilt \(|z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1\) und \(|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1\)

Short answer

a) \(z \in H \Leftrightarrow -1/z \in H\). b) \(|1-z\bar{a}|^2 - |z-a|^2 = (1-|z|^2)(1-|a|^2)\); \(|z| < 1 \iff \left|\frac{z-a}{\bar{a}z-1}\right| < 1\).

Step-by-step solution

Understanding the Half-plane condition

The set \(H\) is defined as the set of complex numbers \(z\) such that the imaginary part of \(z\), denoted as \(\operatorname{Im}(z)\), is greater than 0, i.e., \(H = \{ z \in \mathbb{C}; \operatorname{Im}(z) > 0 \}\). This characterizes the upper half of the complex plane.

Prove \(z \in H \Rightarrow -1/z \in H\)

Assume \(z = x + iy\) where \(x, y \in \mathbb{R}\) and \(y > 0\). Then \(-\frac{1}{z} = -\frac{1}{x+iy} = -\frac{x-iy}{x^2+y^2} = -\frac{x}{x^2+y^2} + i\frac{y}{x^2+y^2}\). The imaginary part of \(-1/z\) is \(\frac{y}{x^2+y^2}\), which is positive since \(y > 0\). Thus, \(-1/z\) is in the upper half-plane \(H\).

Prove \(z \in H \Leftarrow -1/z \in H\)

Assume \(-1/z\) is in \(H\), i.e., the imaginary part of \(-1/z\) is positive. From the previous step, the imaginary part is \(\frac{y}{x^2 + y^2} > 0\), which implies \(y > 0\). Thus, \(z\) is in \(H\). Therefore, \(z \in H \Leftrightarrow -1/z \in H\).

Formulate the expression \(|1-z\bar{a}|^2 - |z-a|^2\)

Use the properties of complex modulus: \(|1-z\bar{a}|^2 = (1-z\bar{a})(1-\bar{z}a)\) and \(|z-a|^2 = (z-a)(\bar{z}-\bar{a})\). Expand both expressions.

Expand \((1-z\bar{a})(1-\bar{z}a)\)

The expansion gives: \(1 - z\bar{a} - \bar{z}a + z\bar{a}\bar{z}a\). Simplify it to \(1 - z\bar{a} - \bar{z}a + |z|^2|a|^2\).

Expand \((z-a)(\bar{z}-\bar{a})\)

The expansion results in: \(|z|^2 - z\bar{a} - \bar{z}a + |a|^2\).

Simplify the difference

Compute the expression \(|1-z\bar{a}|^2 - |z-a|^2\). Substituting the expanded components, the expression simplifies to \(1 - |z|^2|a|^2 - |z|^2 + |a|^2\), which rearranges to \((1-|z|^2)(1-|a|^2)\).

Prove \(|z|

If \(|a| < 1\), use the previous result: \(|1-z\bar{a}|^2 - |z-a|^2 = (1-|z|^2)(1-|a|^2)\). If \(|z| < 1\), it follows from the equation that \(\left|\frac{z-a}{\bar{a}z-1}\right| < 1\). If \(|z| = 1\), then equality holds because the terms in the expansion are exactly zero, \(\left|\frac{z-a}{\bar{a}z-1}\right| = 1\). Utilize algebraic manipulation to establish the equivalence.

Key concepts

Upper Half-Plane

In complex analysis, the upper half-plane refers to the subset of complex numbers where the imaginary part is positive. This can be mathematically represented as \( H = \{ z \in \mathbb{C} ; \operatorname{Im}(z) > 0 \} \). Imagine the complex plane as a two-dimensional space where every point corresponds to a complex number \( z = x + iy \), with \( x \) being the real part and \( y \) being the imaginary part. Thus, the upper half-plane includes all those points above the real axis.

When dealing with problems in complex analysis, recognizing whether a complex number lies in this region can help solve the problem effectively. For instance, in many proofs, such as proving that \( z \in H \) if and only if \( -1/z \in H \), the upper half-plane condition plays a pivotal role. By manipulating and transforming complex numbers, if one proves that the imaginary part of transformed numbers remains positive, it reinforces their position in the upper half-plane.

Complex Modulus

The concept of complex modulus is rooted in the idea of measuring the size or length of a complex number in the complex plane. By formula, the modulus of a complex number \( z = x + iy \) is given by \( |z| = \sqrt{x^2 + y^2} \). It essentially reflects the distance of the point \( z \) from the origin \( (0,0) \).

Understanding complex modulus is key in analyzing properties of circles and disks in the complex plane. For example, if \( |z| < 1 \), the point \( z \) is inside the unit circle. This property is useful when dealing with inequalities and equations involving moduli, such as demonstrating equivalences like \( |z| < 1 \Longleftrightarrow \left| \frac{z-a}{\bar{a}z-1} \right| < 1 \). Calculations involving modulus facilitate understanding how complex numbers behave with respect to each other spatially.

Imaginary Part

The imaginary part of a complex number \( z = x + iy \) is the component \( y \), which gives insight into the behavior of \( z \) on the complex plane. Unlike the real part that determines horizontal placement, the imaginary part determines vertical positioning.

The significance of the imaginary part is highlighted when classifying regions within the complex plane, such as determining whether a number belongs to the upper half-plane. If the imaginary part is greater than zero \( \operatorname{Im}(z) > 0 \), it indicates that \( z \) resides in the upper half-plane. This notion is a cornerstone in understanding mappings and transformations in complex analysis. For instance, the transformation of \( -1/z \) relies on examining the imaginary part to ensure it remains positive, affirming its location within the upper half-plane.

Complex Numbers

Complex numbers are pivotal in extending the concept of numbers beyond the real line. Defined as \( z = x + iy \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit \( i^2 = -1 \), these numbers allow for greater mathematical exploration and expression.

The dual-part structure of complex numbers enables intricate operations and analysis. They are not only defined by their magnitude but also by their direction in the complex plane. Exercises involving complex numbers often require transforming and comparing them, leveraging both their real and imaginary components. For example, the equivalence \( |z| < 1 \) connected to transformations highlights their interaction and position relative to the unit circle. Understanding complex numbers and how they relate to operations like inversion or conjugation can unlock deeper comprehension in complex analysis and its applications.