Problem 15 - 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 \(w_{3}\). \(1, i,-i\) to \(-1,0,3\)

Step-by-Step Solution

Answer

The transformation is \(f(z) = \frac{iz + i}{iz + 1}\).

Step 1: Understanding Linear Fractional Transformation

A linear fractional transformation (LFT), also known as a Möbius transformation, is of the form \( f(z) = \frac{az + b}{cz + d} \). We need to determine the coefficients \(a, b, c,\) and \(d\) such that the transformation maps specific points \(z_1, z_2, z_3\) to \(w_1, w_2, w_3\).

Step 2: Setting Initial Conditions

We have \( f(1) = -1 \), \( f(i) = 0 \), and \( f(-i) = 3 \). These give us three equations with four unknowns. Since linear fractional transformations have one degree of freedom, we use these conditions to solve for the coefficients, assuming one variable (such as \(d\)) and solving for others.

Step 3: Solving for Coefficients

Substitute the conditions into the transformation: 1. \( \frac{a + b}{c + d} = -1 \)2. \( \frac{ai + b}{ci + d} = 0 \) implies \( ai + b = 0 \), thus \( b = -ai \)3. \( \frac{-ai + b}{-ci + d} = 3 \), substituting \(b = -ai\), gives \( \frac{-ai - ai}{-ci + d} = 3 \).From equation 2, \(b = -ai\). Simplifying equation 3, we get \(\frac{-2ai}{-ci + d} = 3\).

Step 4: Simplifying the Equations and Solving

Solve \( \frac{a + b}{c + d} = -1 \) knowing \(b = -ai\). It becomes \( a - ai = -c - d \). Consider solving for each coefficient by assuming \(d = 1\). The system simplifies to three equations:1. \( a + b = -(c + 1) \)2. \( ai = -b \), so \( a = 0 \) and \( b = i \)3. \(\frac{2i}{ci - 1} = -3\), solve for \(c\).

Step 5: Verifying the Outputs

Substitute \(a = 0\), \(b = i\) back into the equations, solving for \(c\) by setting \(\frac{2i}{ci - 1} = 3\) which implies \(2i = 3(ci - 1)\). Solving gives \(c = \frac{2i + 3}{3i} = i\). As \(c\) is solved, verify: if \(f(z)\) indeed maps \(1\), \(i\), \(-i\) appropriately into the expected outputs using these \(a, b, c, d\) values.

Step 6: Constructing the Transformation

Finally, construct the LFT as: \[ f(z) = \frac{iz + i}{iz + 1} \], which maps the points as required.

Key Concepts

Möbius Transformation

The Möbius transformation, also known as the linear fractional transformation, is a fascinating object in mathematics. It takes the form \( f(z) = \frac{az + b}{cz + d} \). This form allows it to map circles and lines in the complex plane to other circles and lines. Möbius transformations are particularly elegant due to their ability to map three distinct points in the complex plane to any other three distinct points. This is done using the cross-ratio, a preserved quantity under Möbius transformations.

To work with Möbius transformations, you typically need to identify the coefficients \( a, b, c, \) and \( d \) that will achieve the desired mapping. This requires setting initial conditions based on known mappings of specific points. For instance, as seen in the exercise, points like 1, \( i \), and \( -i \) are mapped to -1, 0, and 3, respectively. By solving a system of equations, these mappings inform you how to adjust your transformation via the coefficients to get the desired results.

Möbius transformations are foundational in complex analysis because they elegantly combine algebraic, geometric, and analytical aspects, making them very versatile tools.

Complex Analysis

Complex analysis is a branch of mathematics that studies functions of complex numbers. It deals extensively with complex functions, where both the input and output can be complex numbers. This field is especially powerful because it can explore properties and behaviors of entire functions in the complex plane.

A unique feature of complex analysis is its ability to offer insight into the geometric properties of functions. Through transformations like the Möbius transformation, complex analysis provides ways to manipulate and explore complex planes. These transformations can rotate, translate, and even invert the complex plane while preserving angles. This makes complex analysis crucial for various applications, from signal processing to fluid dynamics.

In our exercise, complex analysis helps to map and understand how simple geometric transformations can be used to map specific complex numbers to new positions. It offers a framework whereby seemingly difficult problems can be made more approachable through insights offered by these transformations. By reducing problems to the study of basic transformations, complex analysis reveals deeper symmetries and properties of the complex plane.

Mathematical Mapping

Mathematical mapping refers to the process of associating each element of a set with an element of another set. In mathematics, particularly in the context of complex analysis, mappings often involve functions that transform inputs in a structured and predictable manner.

In our context, the linear fractional transformation is a specific type of mapping employed to transform specific points \( z_1, z_2, z_3 \) to \( w_1, w_2, w_3 \). This type of mapping isn't just a simple translation or rotation. Instead, it's more nuanced. It can be a combination of different transformations that compress, stretch, or rotate the input in intricate ways.

Mathematical mappings like the Möbius transformation play a crucial role in not just transforming points but also in providing a method to understand the underlying structure of the complex plane. These mappings form the basis for more advanced studies and applications across various fields of science and engineering. By understanding how different mappings work, one gains insights into the symmetry and structure of the mathematical landscapes being studied.