Problem 14 - Advanced Engineering Mathematics

Question

A region \(R\) in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image region \(R^{\prime}\) in the \(w\) -plane. Rectangle \(0 \leq x \leq 1,0 \leq y \leq \pi\), under \(w=e^{2}\)

Step-by-Step Solution

Answer

The image region \( R' \) is the single point \( w = e^2 \).

Step 1: Understand the Mapping Function

The given function is a constant map, where every complex number \( z \) in region \( R \) is mapped to the same complex number \( w = e^2 \). This means that irrespective of the input within the defined region, the output will always be \( e^2 \).

Step 2: Define the Image Region

Since the mapping is constant (all values map to the same number), the image region \( R' \) in the \( w \)-plane is simply a single point: \( w = e^2 \).

Step 3: Conclusion of the Image Region

Based on the constant mapping, the region \( R' \) is reduced to just the point \( e^2 \) in the \( w \)-plane, regardless of the shape and bounds of the original rectangle in the \( z \)-plane.

Key Concepts

Constant Mapping

In complex analysis, a **constant mapping** is a transformation where every point in a designated region of the complex plane is mapped to the same single point in another plane. In simpler terms, no matter which point from the original region you choose, the result of applying the mapping function will always be the same specific constant value.

This function is particularly straightforward because it does not depend on the variables of the input points. Consider the function we discussed: if the mapping function is given as \( w = e^2 \), then every point \( z \) within the original region will be mapped exactly to \( e^2 \) in the image plane.

+ **No variation:** The function does not change across different points in the region.+ **Simplification:** Since all points map to the same output, the complexity of the mapping is significantly reduced.In essence, constant mappings take every point in a chosen input area and project it to a single, unvarying point, which simplifies the understanding of the image pattern. This concept widely simplifies many problems in complex mappings where variations would otherwise need to be individually calculated.

Image Region

When considering transformations in the complex plane, the **image region** refers to the set of all points that result from mapping every point in the original region through a given function.

In the case of constant mappings, as specified in our example, the image region simplifies dramatically. Instead of producing a diverse or complex image in the new plane, every input is transformed to a single output point. Specifically, for the constant map \( w = e^2 \), the image region consists solely of the point at \( e^2 \) in the transformed space.

+ **Single Point Image:** The entire region in the original complex plane is compressed into a singular location.+ **Loss of Original Shape:** Any original shapes or boundaries become irrelevant in the image region since they are no longer preserved or visible.This characteristic of having a single point as the image highlights the inherent nature of constant mappings in transforming complex regions into simple outcomes, thereby eliminating the spatial extents of the initial shapes.

Complex Plane

The **complex plane** is a fundamental concept in complex analysis used to graphically represent complex numbers. Each complex number is depicted as a point on this plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

Every complex number can be expressed in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Hence, a point \( (a, b) \) on the complex plane represents the number \( z \).

+ **Real Axis:** Horizontal line for displaying real components.+ **Imaginary Axis:** Vertical line for showing imaginary components.+ **Visual Representation:** Offers a graphical framework for understanding algebraic operations on complex numbers as geometric transformations.In the context of mapping, the complex plane serves as both the domain \( z \)-plane where the original inputs reside and the target \( w \)-plane to which these inputs are transformed. Understanding positions and transformations within this plane is key to grasping complex mappings like the constant map in the problem, where spatial changes are analyzed via these visual coordinates.