Problem 4 - Advanced Engineering Mathematics

Question

In Problems 1-6, find the image of the given line under the mapping \(f(z)=z^{2}\). $$ y=0 $$

Step-by-Step Solution

Answer

The image of the line \(y=0\) under \(f(z)=z^2\) is the non-negative real axis \([0, \infty)\).

Step 1: Understand the Problem

We are given a line defined by the equation \(y=0\), which is the x-axis in the complex plane. We need to find the image of this line under the mapping \(f(z)=z^2\).

Step 2: Express Points on the Line

We express any point on the line \(y=0\) as a complex number where the imaginary part is zero. Therefore, a point on the line is \(z = x + 0i = x\), where \(x\) is a real number.

Step 3: Apply the Mapping

Apply the mapping \(f(z) = z^2\) to the points on the line. This means we replace \(z\) with \(x\) in the expression for \(f(z)\).

Step 4: Calculate the Image

Since \(z = x\), the map becomes \(f(x) = x^2\). By squaring any real number \(x\), the imaginary part remains zero, as \((x)^2 = x^2 + 0i\). Thus, the image is the real line, where each \(x\) on \(y = 0\) maps to \(x^2\) on \(y=0\).

Step 5: Determine the Range of the Image

The range of values for \(x^2\) is from \(0\) to \(\infty)\) for all real numbers \(x\). Therefore, the image is the non-negative real axis, \([0, \infty)\), on the real line.

Key Concepts

Complex Mapping

In the world of complex analysis, a complex mapping can be thought of as a transformation of the complex plane. Mapping functions take each point from one complex space and transform it into another space. A common type of mapping is when a function takes a complex number and produces another complex number. In our exercise, for example, we use the mapping function \(f(z) = z^2\).

• Each input \(z\) is a point on the complex plane represented by \(x+yi\), where \(x\) and \(y\) are real numbers, and \(yi\) is the imaginary part.
• The operation \(z^2\) transforms this point by "squaring" the complex number.
• For the specific problem, since \(y=0\), we simplify \(z\) into just \(x\).

The result is always a new point on the complex plane, which gives us a visualization of how the entire plane can act when this transformation is applied. Complex mappings like these show how shapes and lines in the complex plane change under various transformations, offering deep insights into the geometry of complex numbers.

Complex Plane

The complex plane is a two-dimensional space where every point corresponds to a complex number. This plane is pivotal in complex analysis, bridging the gap between the imaginaries and reals.

• The horizontal axis, also known as the real axis, represents all real numbers.
• The vertical axis, called the imaginary axis, represents all imaginary numbers \(yi\).

Points in this plane are written as \(z = x + yi\). In our exercise, the line \(y=0\) lies along the real axis because the imaginary component is zero.By visualizing complex numbers as points on this plane, complex transformations become easier to comprehend. Therefore, a mapping like \(f(z) = z^2\) can be seen as translating and rotating points around the origin, reshaping lines and curves as they transform. Understanding how points and lines move in the complex plane helps in dissecting complex functions and their behaviors.

Imaginary Part

The imaginary part of a complex number is what makes complex numbers unique. It is denoted by \(i\) and is a critical component of the complex plane.

• A complex number is structured as \(z = x + yi\), where \(x\) is the real part, and \(yi\) is the imaginary part.
• The imaginary part is zero on the real axis, which is the line we are dealing with in our exercise.

Even though the imaginary part is zero in our specific problem, its general concept is essential in understanding how complex numbers behave. In transformations or mappings, the imaginary part can affect both the magnitude and direction of the transformation. Knowing how the imaginary part influences a complex mapping can give greater insight into the final position and shape of the image on the complex plane.

Real Axis

The real axis on the complex plane is a concept that many are already familiar with—it works just like the number line from real number systems, but it is a vital component of the complex plane. In this plane:

• The real axis is where all imaginary parts of complex numbers are zero.
• It stretches horizontally, meaning all points on it have only real values.

In the exercise at hand, the line \(y=0\) is equivalent to this real axis. When mapped by the function \(f(z) = z^2\), each point \(x\) along this axis transforms into \(x^2\). The image remains on the real axis, but only on the non-negative portion, reflecting how real parts can influence transformations purely on their real values.Understanding the behavior of the real axis in complex mappings allows for an intuitive grasp of linear transformations and the resulting outputs on the complex plane.