Problem 10 - Advanced Engineering Mathematics

Question

Construct a linear fractional transformation that maps the given triple \(z_{1}, z_{2}\), and \(z_{3}\) to the triple \(w_{1}, w_{2}\), and \(w_{3}\). \(i, 0,-i\) to \(0,1, \infty\)

Step-by-Step Solution

Answer

The transformation is \( f(z) = \frac{z - i}{-z - i} \).

Step 1: Identify Given Points and Map Points

We have the points: - The given triple to map from: - \( z_1 = i \) - \( z_2 = 0 \) - \( z_3 = -i \)- The triple to map to: - \( w_1 = 0 \) - \( w_2 = 1 \) - \( w_3 = \infty \)

Step 2: Construct the General Form of a Linear Fractional Transformation

A linear fractional transformation is given by:\[ f(z) = \frac{az + b}{cz + d} \]We need to find the constants \(a, b, c, d\) such that the mapping holds for our given sets of points.

Step 3: Apply Conditions to Determine Coefficients

Substitute the given points to find coefficients:1. For \(f(z_1 = i) = 0\), we get \(ai + b = 0\)2. For \(f(z_2 = 0) = 1\), we get \(b/d = 1\)3. For \(f(z_3 = -i) \rightarrow \infty\), we require \(c(-i) + d = 0\)

Step 4: Solve System of Equations

From the given conditions:- From condition 1: \(b = -ai\)- From condition 2: \(b = d\)- From condition 3: \(d = ci\)Substitute \(b = -ai\) into \(b = d\), obtain \(d = -ai\). Substitute \(d = ci\) to get \(-ai = ci\) hence \(c = -a\).

Step 5: Set a Parameter and Solve

Let \(a = 1\): - Then \(c = -1\) - From \(b = -ai\), we get \(b = -i\)- From \(d = -ai\), we get \(d = -i\)The transformation is:\[ f(z) = \frac{z - i}{-z - i} \]

Step 6: Verify the Transformation

Verify each point:- For \(z = i\), \( f(i) = \frac{i-i}{-i-i} = 0\)- For \(z = 0\), \( f(0) = \frac{0-i}{0-i} = rac{-i}{-i} = 1\)- For \(z = -i\), \( f(-i) = \frac{-i-i}{i-i} = \text{undefined, maps to } \infty\)The mapping from \(i, 0, -i\) to \(0, 1, \infty\) is correct.

Key Concepts

Mapping

Mapping in mathematics refers to the process of transforming a set of numbers or functions into another set. This can be as simple as translating points to a new coordinate or as complex as turning functions into different forms.

In the context of linear fractional transformations, which are a type of complex function mapping, we take a specific set of complex numbers and transform them into another target set. The goal is to preserve the structure of the original set while transitioning it to a different representation.

In our exercise, we aimed to map the points - - **i**, - **0**, - **-i** - to the points - **0**, - **1**, - **∞** Applying the linear fractional transformation ensures that these original points are exactly represented as new target points, demonstrating the power and flexibility of mapping in complex analysis.

Complex Analysis

Complex Analysis explores functions of complex numbers, focusing on properties that differ from those of real numbers. It delves into complex number operations and the unique aspects when plotted on the complex plane.

One significant aspect of complex analysis is the study of linear fractional transformations. These transformations allow any three distinct points in the complex plane to be mapped to any other three points. This versatility originates from the intricate nature of the complex number system. Linear fractional transformations take the form:\[ f(z) = \frac{az + b}{cz + d} \]This formula uses constants \(a, b, c,\) and \(d\) from complex numbers, maintaining intricate relationships between affine transformations and Möbius transformations.

The use of complex analysis in understanding such mappings enables mathematicians to simplify, transform, and visualize complex relationships, making it a vital tool in both theoretical and applied mathematics.

Problem Solving

Problem solving in mathematics involves finding solutions to well-defined problems using logical reasoning and known methods. In exercises like the one discussed, this process is broken into a step-by-step approach to ease comprehension.

For the problem of mapping points via a linear fractional transformation, we identified the given and target points, ensuring we understand the foundational requirements of this transformation. We then constructed a general form, applied conditions using our specific points, and solved resulting equations. This systematic approach made it easier to uncover the values of the constants in the transformation formula.

This methodical approach is invaluable in identifying errors early, verifying results, and gaining deeper insights into the underlying mathematical principles. It also fosters critical thinking and logical deduction skills that are beneficial both academically and in real-world applications.

Mathematics Education

Mathematics education emphasizes understanding concepts thoroughly and applying them to solve various problems. Linear fractional transformations are crucial in the curriculum as they highlight connections between algebra and geometry in the realm of complex numbers.

By dissecting problems into smaller steps, like mapping points with these transformations, students learn to handle complex tasks with ease and confidence. Educational approaches often stress the importance of visualizing problems, using technology and graphical representations to aid comprehension, and practice through repetition to solidify understanding.

Moreover, enriching textbooks or exercise solutions with detailed explanations and guiding questions supports students in independent learning. This fosters a deep comprehension that extends beyond rote memorization, enabling students to cross-apply their knowledge across different areas of mathematics and to solve problems innovatively.