Problem 28 - Advanced Engineering Mathematics

Question

The power function \(z_{1}=z^{2 / 3}\) maps the wedge \(0 \leq \operatorname{Arg} z \leq 3 \pi / 2\) to the upper half-plane \(0 \leq \operatorname{Arg} z_{1} \leq \pi .\) We then let \(w=e^{-i \pi / 2} z_{1}+2=-i z_{1}+2\) to rotate the upper half-plane through \(-90^{\circ}\) and then translate 2 units to the right. Therefore the composite function is \(w=-i z^{2 / 3}+2\) and the image region is the half-plane \(u \geq 2.\)

Step-by-Step Solution

Answer

The composite function \(w = -i z^{2/3} + 2\) maps wedge \(0 \leq \text{Arg} \ z \leq \frac{3\pi}{2}\) to the half-plane \(u \geq 2\).

Step 1: Identify the Complex Function

The exercise provides the power function \( z_1 = z^{2/3} \). This function takes a complex number \( z \) and raises it to the power of \( \frac{2}{3} \). The input region is the wedge defined by the argument range \( 0 \leq \text{Arg} \ z \leq \frac{3\pi}{2} \).

Step 2: Determine Image Under Power Function

The power function \( z_1 = z^{2/3} \) maps the wedge \(0 \leq \text{Arg} \ z \leq \frac{3\pi}{2} \) onto the region \(0 \leq \text{Arg} \ z_1 \leq \pi \). This transformation essentially compresses the wedge into the upper half-plane of the complex plane, bounded by real values.

Step 3: Apply Rotation Transformation

Define \( w = e^{-i\pi/2} z_1 + 2 = -i z_1 + 2 \) as a transformation on \( z_1 \). Here, multiplying by \(-i\) rotates \( z_1 \) by \(-90^{\circ} \), effectively reflecting it across the real axis.

Step 4: Translate the Image

Continuing with \( w = -i z_1 + 2 \), the addition of 2 translates the rotated image 2 units to the right. Consequently, the image of the upper half-plane \( 0 \leq \text{Arg} \ z_1 \leq \pi \) is shifted to \( u \geq 2 \), where \( u \) is the real part of \( w \).

Step 5: Final Result of the Transformation

The composite function \( w = -i z^{2/3} + 2 \) transforms the original wedge to the half-plane \( u \geq 2 \). This describes the entire process starting from \( z \) and ending with the transformed region in terms of \( w \).

Key Concepts

Power Function

A power function in complex analysis means taking a complex number and raising it to a specific power. In this exercise, we have the function \( z^{2/3} \). Here, the complex number \( z \) is raised to the fractional power \( \frac{2}{3} \). This involves both changing the magnitude of \( z \) and modifying its angle, or argument, on the complex plane.
The power function has a transformative effect:

• The magnitude of \( z \) raised to \( \frac{2}{3} \) is \(|z|^{2/3}\), meaning it's the cube root of the square of the magnitude.
• The argument of \( z \), \( \text{Arg} \, z \), is multiplied by \( \frac{2}{3} \). If \( \text{Arg} \, z \) is initially in the range of \( 0 \leq \text{Arg} \, z \leq \frac{3\pi}{2} \), the result is compressed into a smaller range, effectively mapping out a different area in the complex plane.

This transformation helps in visualizing complex structures and their interactions within designated regions of the complex plane.

Complex Transformation

Complex transformations involve changing the position and form of regions on the complex plane using certain functions. In this exercise, the complex transformation is performed using the function \( w = -i z^{2/3} + 2 \).
Let's break down what's happening:

• First, there's a mapping of \( z \) using \( z^{2/3} \), which shifts and changes the original region defined by the wedge.
• Then, by applying \( -i \), we rotate the transformed region by \(-90^{\circ}\). This rotation changes the orientation of the region on the complex plane by reflecting it.
• Finally, we add 2, translating the entire region two units to the right. This translation affects the horizontal position, altering the real part of every point.

The synthesis of these steps results in a transformative pathway from one part of the complex plane to another, helping to better understand and manipulate complex numbers and regions.

Argument of Complex Number

The argument of a complex number is an essential part of understanding its location and orientation on the complex plane. The argument, denoted as \( \text{Arg} \, z \), represents the angle formed between the positive real axis and the line representing the complex number.
For example:

• In the initial region, the argument of \( z \) is within \( 0 \leq \text{Arg} \, z \leq \frac{3\pi}{2} \). This range covers a wedge, nearly wrapping around three-quarters of a circle.
• When transformed using \( z^{2/3} \), the argument is multiplied by \( \frac{2}{3} \), mapping it to a new range: \( 0 \leq \text{Arg} \, z_1 \leq \pi \).

This transformation compresses the argument range from a broad wedge to a half-plane, signifying how power functions can concentrate or expand angles on the complex plane.

Complex Rotation

In complex analysis, rotating a complex number changes its angle, or argument, while keeping its magnitude fixed. The rotation in this problem is performed by multiplying with \(-i\), which rotates the argument of the complex number by \(-90^{\circ}\).
This operation is crucial:

• Without changing the magnitude, \(-i\) moves every point counterclockwise on the plane relative to the origin.
• Since \(-i\) translates the upper half-plane into another half-plane that opens to the right of the imaginary axis, it realigns how we view the complex number structure.

Effectively, the rotation reorients regions on the complex plane, showing how mathematical operations transform spatial relationships and provide a new perspective on data within complex analysis.