Problem 14 - 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 \(w_{3}\). \(-1,0,1\) to \(\infty,-i, 1\)

Step-by-Step Solution

Answer

The transformation is \( f(z) = \frac{(2+i)z - i}{z + 1} \).

Step 1: Understanding the Linear Fractional Transformation

A linear fractional transformation is a function of the form \( f(z) = \frac{az + b}{cz + d} \) where \( a, b, c, \) and \( d \) are complex numbers and \( ad - bc eq 0 \). This transformation maps one set of points in the complex plane to another.

Step 2: Setting Up the Transformation Conditions

To determine the transformation, you need to satisfy the conditions that the points \(-1, 0, \) and \( 1 \) are mapped to \( \infty, -i, \) and \( 1 \) respectively. Therefore:\(\begin{align*}f(-1) & = \infty, \f(0) & = -i, \f(1) & = 1.\end{align*}\)

Step 3: Apply the Condition for \(z_1 = -1\)

If \( f(-1) = \infty \), then \( -c + d = 0 \). This implies that \( d = c \).

Step 4: Apply the Condition for \(z_2 = 0\)

Since \( f(0) = -i \), we have \( f(0) = \frac{b}{d} = -i \). Thus, \( b = -id \).

Step 5: Apply the Condition for \(z_3 = 1\)

As \( f(1) = 1 \), we equate \( \frac{a + b}{c + d} = 1 \). Substituting \( b = -id \) and \( d = c \), this becomes:\[ a - id = 2c. \]

Step 6: Solve the System of Equations

We now have the system:\(\begin{align*}d &= c, \b &= -id, \a - id &= 2c.\end{align*}\)Substitute \( b \) and \( d \):\( a - i(c) = 2c \) leads to \( a = 2c + ic \) which simplifies to \( a = (2 + i)c \).

Step 7: Determining the Transformation Coefficients

Let \( c = 1 \), a possible choice to satisfy \( a = (2+i)c \). This gives us:\[ a = 2 + i, \quad b = -i, \quad c = 1, \quad d = 1. \]

Step 8: Construct the Transformation

Substitute \( a, b, c, \) and \( d \) into the transformation function:\[ f(z) = \frac{(2+i)z - i}{z + 1}. \]

Step 9: Verify the Mapping

Check each mapping condition with the function:- For \( z = -1 \), \( f(-1) = \frac{(2+i)(-1) - i}{-1 + 1} = \frac{-\infty}{0} = \infty \).- For \( z = 0 \), \( f(0) = \frac{-i}{1} = -i \).- For \( z = 1 \), \( f(1) = \frac{2+i - i}{1+1} = 1 \). All conditions are satisfied.

Key Concepts

Complex Numbers

Complex numbers are a foundational element in mathematics, particularly when dealing with transformations like the linear fractional transformation. A complex number is expressed as a combination of a real part and an imaginary part:

• Real part is denoted as a real number, for example, 2.
• Imaginary part is represented as an imaginary unit 'i,' which satisfies the equation \( i^2 = -1 \), for instance, 3i.

Thus, a complex number looks like 2 + 3i. When working with transformations, these numbers serve as coordinates on the complex plane. They help map one set of values onto another using specified rules. Understanding how to manipulate complex numbers is crucial for applying and constructing transformations adequately. This manipulation includes addition, subtraction, multiplication, and division, all of which adhere to standard operations while also considering the unique property that \( i^2 = -1 \).

Mapping Conditions

Mapping conditions are the desired outcomes for each point transformation in a linear fractional transformation. They specify where each input complex number \( z \) should map under the transformation to a new output number \( w \).
To set up mapping conditions:

• Identify the points in the original complex plane like \( z_1, z_2, \) and \( z_3 \).
• Determine their corresponding locations in the target map like \( w_1, w_2, \) and \( w_3 \).

For instance, mapping points like

• \( z_1 = -1 \) maps to \( w_1 = \infty \)
• \( z_2 = 0 \) maps to \( w_2 = -i \)
• \( z_3 = 1 \) maps to \( w_3 = 1 \)

This understanding is essential because each condition becomes an equation in your system, helping you find the coefficients of your transformation function. These conditions ensure that the transformation matches the specified mapping.

Transformation Function

A transformation function is a type of mapping, specifically a linear fractional transformation in this context, which is represented by the formula \( f(z) = \frac{az + b}{cz + d} \). This formula transforms a complex input \( z \) to another complex number. Key components include:

• Coefficients \( a, b, c, \) and \( d \) are complex numbers.
• A requirement is that \( ad - bc eq 0 \) to ensure the function is well-defined and has an inverse.

The transformation function is utilized to map each input \( z \) to a pre-determined output \( w \) using the transformation's coefficients. To find these coefficients, one must solve the system of equations derived from the mapping conditions set for the specific transformation task. When the transformation is accurately defined, every input-output pair should satisfy the transformation function.

System of Equations

Creating and solving a system of equations is crucial in the context of determining the coefficients for the linear fractional transformation. Based on the mapping conditions, each condition translates to an equation that relates to the coefficients \( a, b, c, \) and \( d \). For instance, given conditions:

• If \( f(-1) = \infty \), it means \(-c + d = 0 \).
• If \( f(0) = -i \), it implies that \( \frac{b}{d} = -i \).
• If \( f(1) = 1 \), you have \( \frac{a+b}{c+d} = 1 \).

These conditions bring together equations, such as \(-c + d = 0\), \( b = -id\), and \( a - id = 2c \), creating a system. Solving this system enables you to calculate the specific values for \( a, b, c, \) and \( d \), which make the transformation function valid. Once solved, these values are substituted back into the transformation equation, allowing the satisfaction of all given mapping conditions.