Problem 9 - Complex Analysis

Question

Which of the following domains are star-shaped? (a) \(\\{z \in \mathbb{C} ; \quad|z|<1\) and \(|z+1|>\sqrt{2}\\}\) (b) \(\\{z \in \mathbb{C} ; \quad|z|<1\) and \(|z-2|>\sqrt{5}\\}\), (c) \(\\{z \in \mathbb{C} ; \quad|z|<2\) and \(|z+\mathrm{i}|>2\\}\) In each case determine the set of all star centers.

Step-by-Step Solution

Answer

None of the domains (a), (b), or (c) are star-shaped; they have no star centers.

Step 1: Understand Star-Shaped Domains

A set \( S \) in the complex plane \( \mathbb{C} \) is star-shaped if there exists a point \( z_0 \in S \) such that for every point \( z \) in \( S \), the line segment from \( z_0 \) to \( z \) is entirely contained in \( S \). The point \( z_0 \) is called a star center.

Step 2: Analyze Domain (a)

Consider the set \( S_a = \{ z \in \mathbb{C} ; |z|<1 \text{ and } |z+1|>\sqrt{2} \} \). This domain is the intersection of a disk of radius 1 centered at the origin and the exterior of another disk of radius \(\sqrt{2}\) centered at \(-1\). Geometrically, the first disk includes the point \(-1\) and the second disk excludes the immediate surrounding area around \(-1\). As the intersection does not cover the line segment between some potential center such as \(-1\) and other points like 0 or near the boundary of the first disk, no line segment can be fully inside both disks, implying that this domain is not star-shaped.

Step 3: Analyze Domain (b)

Consider the set \( S_b = \{ z \in \mathbb{C} ; |z|<1 \text{ and } |z-2|>\sqrt{5} \} \). This domain is the intersection of a disk of radius 1 centered at the origin and the exterior of a disk of radius \( \sqrt{5} \) centered at \( 2 \). Since the first disk includes the origin and stays quite far from the center of the second disk, no part of this domain satisfies the connectivity requirement for a line segment entirely inside it. Thus, \( S_b \) is not star-shaped.

Step 4: Analyze Domain (c)

Consider the set \( S_c = \{ z \in \mathbb{C} ; |z|<2 \text{ and } |z+\mathrm{i}|>2 \} \). This domain is the intersection of a disk of radius 2 centered at the origin and the exterior of a disk radius 2 centered at \(-i\). The intersection forms a crescent shape, and one point near the real axis can be chosen such that the segment from this point to any point within the crescent will remain entirely inside the crescent. However, the point cannot simply be chosen inside the interior disk anywhere near the upper half of the plane. In fact, no such singular point exists for this domain. Thus, the domain is not star-shaped either.

Step 5: Conclusion: Evaluate Star Centers

Based on each analysis, there are no star centers for any of the domains (a), (b), and (c). Each domain lacks a point where a line segment can be entirely inside the respective region from the potential center to every other point.

Key Concepts

Complex Plane

The complex plane is a visual and mathematical way to represent complex numbers.
Every complex number can be written in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
The complex plane consists of a horizontal axis, known as the real axis, which represents the real part of the complex numbers, and a vertical axis, known as the imaginary axis, which represents the imaginary part.

In the complex plane, you can visualize complex numbers as points or vectors. This graphical representation allows for easy geometric interpretations of operations like addition, subtraction, and multiplication of complex numbers.
This becomes essential when analyzing shapes, such as disks and domains, within this plane.

• Real part represented along the x-axis.
• Imaginary part represented along the y-axis.
• Each point describes a unique complex number.

Understanding this allows us to explore and define concepts like star-shaped domains within the complex plane.

Line Segments

A line segment in the complex plane is simply the shortest path connecting two points.
In geometric terms, if you have two complex numbers, say \( z_0 = x_0 + y_0i \) and \( z_1 = x_1 + y_1i \), the line segment connecting \( z_0 \) and \( z_1 \) consists of all points \( z(t) \) of the form:
\[z(t) = (1-t)z_0 + tz_1 = ((1-t)x_0 + tx_1) + ((1-t)y_0 + ty_1)i, \ \ 0 \leq t \leq 1\]
When determining if a domain in the complex plane is star-shaped, we look for a point \( z_0 \) such that every line segment connecting \( z_0 \) to any other point in the set lies entirely within the domain.

• Defined by endpoints.
• Use parameter \( t \) to represent points on the segment.
• Test line segments for inclusion within domains to check star-shaped property.

Mastering the concept of line segments is crucial to understanding properties of complex domains.

Intersection of Disks

Intersections of disks in the complex plane can form intricate shapes.
To visualize this, consider a disk centered at a point in the complex plane, having a specified radius.
In mathematical terms, the disk centered at \( z_c \) with radius \( r \) is the set of all points \( z \) such that \(|z - z_c| < r\).
When two disks intersect, their overlapping region defines a new shape. This intersection is crucial when analyzing specific domains for properties like being star-shaped.

For solving our exercises, intersections involve:

• Intersection of a disk and an exterior of another disk.
• Understanding where the overlap shapes the domain.
• Using inequalities to define and understand these regions.

The geometry and math combined help determine possible paths within the intersection.

Star Centers

A star center of a domain is a special point.
It is a point from which a line drawn to any other point within the domain remains entirely inside the domain.
This makes the investigation of star centers central to understanding star-shaped regions.
If you cannot find such a point in a given domain, then the domain is not star-shaped.

Star centers illustrate:

• The ability of a single point to maintain connectivity throughout the region.
• The specific geometric and algebraic setup that allows or disallows finding such a point.
• In our examples, no such point exists for any of the domains analyzed, illustrating that none are star-shaped.

Studying star centers helps in understanding complex shapes and connectivity better.