Problem 1 - Advanced Engineering Mathematics

Question

A curve in the \(z\) -plane and a complex mapping \(w=f(z)\) are given. In each case, find the image curve in the \(w\) -plane. \(y=x\) under \(w=1 / z\)

Step-by-Step Solution

Answer

The image curve is a rectangular hyperbola centered at the origin in the w-plane.

Step 1: Express z in Terms of x and y

The line \(y = x\) in the \(z\)-plane means that every point on the line satisfies this equation. In terms of complex numbers \( z = x + iy \), this simplifies to \( z = x + ix = x(1 + i) \).

Step 2: Determine w in terms of x

The mapping is given by \( w = \frac{1}{z} \). Substitute \( z = x(1 + i) \) to get \( w = \frac{1}{x(1+i)} \).

Step 3: Simplify the Expression for w

To simplify \( w = \frac{1}{x(1+i)} \), multiply the numerator and the denominator by \(1 - i\) (the conjugate of \(1+i\)): \[ w = \frac{1}{x(1+i)} \cdot \frac{1-i}{1-i} = \frac{1-i}{x((1+i)(1-i))} \]. Since \((1+i)(1-i) = 1^2 - i^2 = 2\), this simplifies to \( w = \frac{1-i}{2x} \).

Step 4: Analyze the Image Curve

Since \(w = \frac{1-i}{2x}\), it can be expressed as \( w = \frac{1}{2x} - i\frac{1}{2x} \). As \(x\) varies from \(-\infty \) to \(\infty\), the real part \( \frac{1}{2x} \) and the imaginary part \( -\frac{1}{2x} \) are inversely related, forming a rectangular hyperbola in the \(w\)-plane with vertical and horizontal asymptotes at (0, 0).

Step 5: Identify the Quadrants and End Behavior

As \(x\) increases positively, \( \frac{1}{2x} \) and \(-\frac{1}{2x}\) get smaller (approach zero), indicating that the curve approaches the origin. When \(x\) approaches zero from both sides, \( \frac{1}{2x}\) and \(-\frac{1}{2x}\) become very large, implying that the curve moves away infinitely. For negative \(x\), the real and imaginary components swap signs, but the same hyperbolic shape is retained, symmetrically across the origin.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. In mathematical notation, a complex number is often expressed as \( z = x + iy \), where \( x \) is the real part, and \( y \) is the imaginary part, represented as \( i \times y \). Here, \( i \) is the imaginary unit with the property \( i^2 = -1 \).
Using complex numbers can greatly simplify many mathematical problems, especially those involving geometry and mapping in the complex plane.

• The real part \( x \) determines the position along the horizontal axis.
• The imaginary part \( y \) determines the position along the vertical axis.

This results in complex numbers being represented as points in a plane, known as the complex plane or \( z \)-plane. This concept is key to understanding complex mappings, as transformations in this plane can provide visual interpretations of equations and functions.

Rectangular Hyperbola

A rectangular hyperbola is a special type of curve that can be defined by an inverse relationship between two variables. In the context of our exercise, we see a rectangular hyperbola emerge in the \( w \)-plane. This is due to the presence of terms in \( w = \frac{1-i}{2x} \) that inversely relate the real and imaginary components.

• Rectangular hyperbolas have the characteristic shape of two symmetrical curves extending to infinity in two opposite quadrants.
• The basic form of a rectangular hyperbola can be given by \( xy = c^2 \), where \( c \) is a constant. This indicates that as one variable increases, the other decreases proportionally.

In complex mappings, recognizing a rectangular hyperbola helps us visualize the transformation and understand the behavior of the mapped curve in the complex plane.

Asymptotes

Asymptotes are lines that a curve approaches closer and closer but never actually touches. In the \( w \)-plane, our rectangular hyperbola is associated with asymptotes at (0, 0). These asymptotes help us understand the end behavior of the curve as the variables involved grow very large or very small.

• Vertical asymptotes can occur when the denominator of a function approaches zero, causing the function value to approach infinity.
• Horizontal asymptotes refer to the behavior of a function as it moves towards vast positive or negative values along the axis.
• In this exercise, the curve becomes closer to the asymptotes as \( x \) tends to either very small or very large magnitudes.

Given that a curve can approach an asymptote without intersecting, understanding these lines provides insight into both the local and global behavior of functions in complex mappings.

Inverse Relationship

An inverse relationship in mathematics occurs when two variables are related such that as one increases, the other decreases. In the given mapping \( w = \frac{1 - i}{2x} \), the real and imaginary parts exhibit this inverse relationship.

• As \( x \) increases, \( \frac{1}{2x} \) decreases, moving the curve towards the origin.
• Conversely, when \( x \) decreases towards zero, \( \frac{1}{2x} \) and \( -\frac{1}{2x} \) become larger, moving the curve away from the origin.

Understanding inverse relationships is crucial in mathematical analysis as they often reveal the symmetrical nature of functions and curves, such as those found in complex mappings. Observing this behavior helps predict the trends and directions in which the curve is likely to evolve, contributing to a more comprehensive understanding of the transformations taking place.