Question

An angular domain in the cemplex plane is defined by \(0

Short answer

The correct option is (ii) \( w = iz^4 \).

Step-by-step solution

Understanding the Mapping of Angular Domains

Any complex number can be expressed as \( z = re^{i\theta} \), where \( r \) is the modulus and \( \theta = \text{arg}(z) \) is the argument. The region defined by \( 0 < \text{arg}(z) < \frac{\pi}{4} \) is an angular domain with angle \( \frac{\pi}{4} \). We need to find the mapping function that transforms this region into the left half-plane (\( \text{Re}(w) < 0 \)).

Applying Transformation to Map the Domain

The transformation that is being applied to \( z \) is \( w = f(z) \). Each option represents a different transformation: - (i) \( w = z^{4} \)- (ii) \( w = iz^{4} \)- (iii) \( w = -z^{4} \)- (iv) \( w = -i z^{4} \)The goal is to identify which transformation maps the original domain (\( 0 < \text{arg}(z) < \frac{\pi}{4} \)) to the left half-plane (where \( \text{Re}(w) < 0 \)).

Checking Each Option

1. **Option (i)**: \( w = z^4 \): - \( \text{arg}(w) = 4\text{arg}(z) = 4\theta \). The maximum \( \text{arg}(w) = 4 \times \frac{\pi}{4} = \pi \). Thus, \( w \) can take values in the entire left half-plane. Incorrect since it includes both left and right half-planes.2. **Option (ii)**: \( w = i z^4 \): - \( \text{arg}(w) = \text{arg}(i) + 4\text{arg}(z) = \frac{\pi}{2} + 4\theta \). The range is \( \frac{\pi}{2} < 4\theta < \frac{\pi}{2} + \pi = \frac{3\pi}{2} \). This correctly maps into the left half-plane.3. **Option (iii)**: \( w = -z^4 \): - \( \text{arg}(w) = \pi + 4\text{arg}(z) \), which maps to the entire left half-plane.4. **Option (iv)**: \( w = -iz^4 \): - \( \text{arg}(w) = -\frac{\pi}{2} + 4\text{arg}(z) \), not mapping into left half-plane exclusively for the given \( z \).

Conclusion and Correct Answer

Option (ii), \( w = iz^{4} \), correctly maps the angular domain \( 0 < \text{arg}(z) < \frac{\pi}{4} \) to the left half-plane, as the resulting argument \( \frac{\pi}{2} < \text{arg}(w) < \frac{3\pi}{2} \) ensures the real part of \( w \) is negative.

Key concepts

Complex Mapping

In complex analysis, complex mappings refer to transformations that alter the structure or properties of complex domains.
A mapping of a complex number, represented as \( z = re^{i\theta} \), involves both the modulus \( r \) and the argument \( \theta \).
This transformation typically is denoted as \( w = f(z) \), where \( w \) is dependent upon \( z \) through a specific function.For instance, transformations can be polynomial, exponential, or trigonometric.
Each transformation affects the complex plane differently. Some turn circles into lines, while others could alter entire domains.
In our exercise, we are looking at transformations that modify the angular domain into the left half-plane.To comprehend these further, knowing the impact of such transformations helps predict how a region in the complex plane alters.
For example, a mapping \( w = z^n \) multiplies the angle by \( n \).
Understanding these fundamental transformations lays the groundwork for more intricate operations in complex analysis.

Angular Domains

Angular domains in the complex plane are specific regions defined by the argument of complex numbers.
When we mention an angular domain like \( 0 < \text{arg}(z) < \frac{\pi}{4} \), we describe a sector originating from the origin and extending outward.This domain specifies a range of angles measured counterclockwise from the positive real axis.
It forms a wedge-like region of the complex plane.

• This region is significant in mapping as it dictates which transformation will map it into another desired region such as the left half-plane.

Understanding angular domains involves recognizing the effect of argument multiplication under transformations.
Many mapping transformations, like the one in the exercise, depend on this property to determine how they alter the domain.
Grasping how these domains operate provides insights into more complex mappings.

Argument of Complex Numbers

The argument of a complex number is essentially the angle that number makes with the positive real axis.
Represented as \( \text{arg}(z) \), it is crucial in understanding transformations and their implications in complex analysis.In polar form, a complex number \( z \) can be expressed as \( z = re^{i\theta} \).
Here, \( \theta \) represents the argument, dictating the number's direction from the origin.Transformations involving angles will scale or shift these arguments.
For example, in the transformation \( w = z^n \), the argument \( \theta \) is multiplied by \( n \).
This effectively scales it, expanding or contracting the domain in which \( z \) can reside.Understanding and manipulating these arguments help predict how transformation affects particular regions.

• This knowledge is critical in mappings like those transforming angular domains.

Left Half-Plane Transformation

The left half-plane in the complex plane is a region where the real part of any number is negative.
Mathematically, it is defined as \( \text{Re}(w) < 0 \). Transforming a given domain into this space is a common task in complex mapping.The exercise provided showcases a transformation that successfully maps an angular domain onto the left half-plane.

• In this scenario, the transformation chosen is \( w = iz^4 \), one that ensures every part of the angular domain fills the negative real axis side.

This transformation works by adjusting the argument of \( z \) such that the real part of \( w \) becomes negative.
By understanding transformations and their roles, we can effectively choose functions that map domains as required.Grasping these transformations provides the tools to manipulate complex regions confidently.
Consequently, identifying suitable mappings becomes intuitive.