Problem 21 - Advanced Engineering Mathematics

Question

Find a complex mapping from the given region \(R\) in the \(z\) -plane to the image region \(R^{\prime}\) in the \(w\) -plane. Strip \(1 \leq y \leq 4\) to the strip \(0 \leq u \leq 3\)

Step-by-Step Solution

Answer

The mapping is \( w = z - i \), shifting the strip down by one unit on the imaginary axis.

Step 1: Identify the Transformation Type

The problem involves mapping a vertical strip from the complex plane, indicating we likely need a linear transformation. The strip is defined by imaginary parts of complex numbers.

Step 2: Use the General Mapping Formula

A linear complex mapping can typically take the form: \( w = az + b \), where \( w = u + iv \) and \( z = x + iy \). Adjusting such equations could transform vertical strips.

Step 3: Solve for Mapping Constants (Magnitude)

Since the strip is determined by vertical bounds on the imaginary part \( y \), we start with the map \( v = ky + c \). Here, we need to match the size of the original strip (imaginary size = 3) with the new strip size. Set up:\( 1 \to 0 \, \text{and} \, 4 \to 3 \). Due to linearity, solve: \[ v = k(y - 1) \times \left( \frac{3}{3} \right) \rightarrow v = y - 1 \].

Step 4: Set up the Entire Transformation Formula

Combine real and imaginary components to get the entire transformation: \( w = a(x + iy) + b \).For \( a = i \) and \( b = -i \), this yields:\( w = i(y - 1)+i(x) = (x)+(y-1)i \).

Step 5: Validate the Transformation

Confirm: the imaginary part transformation \( y \to y - 1 \) leads to the adjusted limits \( 1 \to 0 \) and \( 4 \to 3 \), conforming to the target strip.

Key Concepts

Linear Transformation

A linear transformation in complex analysis is a function that maps a complex number to another complex number in a specific, consistent way. In the context of the complex plane, this usually means transforming points via an equation, typically written as:

• \( w = az + b \)

where:

• \( w \) is the image (output) point,
• \( z \) is the original (input) point,
• \( a \) and \( b \) are complex numbers that determine the specifics of the transformation.

This formula means the transformation changes both the magnitude and phase of a point in the complex plane. Changing these parameters to achieve a desired mapping—such as mapping a specific strip in the complex plane to another strip—requires solving for \( a \) and \( b \). The task becomes tweaking these values so the transformation smoothly shifts and scales points as needed.

Complex Mapping

Complex mapping refers to applying a specific rule or transformation to every point in a region of the complex plane. This rule moves each point to a corresponding point in a new region. In the exercise, we look at a vertical strip defined by imaginary parts, and the task is to transform it into another strip with possibly different dimensions or orientations.The complex mapping rule for this specific exercise is:

• \( w = az + b \)

To find the constants \( a \) and \( b \), observe how the boundaries of the strip transform. The strip changes from the imaginary interval \( 1 \leq y \leq 4 \) to \( 0 \leq u \leq 3 \). Solving these boundary conditions mathematically helps land on the proper constants to ensure each point shifts correctly.

Complex Plane

The complex plane is like a two-dimensional graph but designed for visualizing complex numbers. In this plane, each number has two components: a real part and an imaginary part. Specifically:

• The horizontal axis represents real numbers.
• The vertical axis represents imaginary numbers.

In our exercise, the complex plane defines our starting strip. Understanding this representation allows for interpreting transformations visually and mathematically. When working with strips (like the one from \( y=1 \) to \( y=4 \)), the focus is mostly on shifting the imaginary component to map onto a new range, aligning neatly with practical applications in fields like electrical engineering and fluid dynamics.

Image Region

The image region is the end destination for a point mapping from the original region. After applying a transformation, every point within the original region has a corresponding point in this new region.In the exercise, the original region is the vertical strip determined by imaginary parts, \( 1 \leq y \leq 4 \). After transformation, the strip changes to \( 0 \leq u \leq 3 \), defined in terms of the real part, \( u \), of \( w = u + iv \). The aim is to understand and ensure each step of this transformation correctly and proportionally adjusts to meet these new boundary conditions.Such transformations and mappings not only help in visualizing complex numbers but are essential in practical scenarios such as signal processing, designing filters, and more. They change how data or signals appear but consistently, preserving essential relationships and patterns.