Problem 10 - 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. Circle \(|z|=1\) under \(w=z+1 / z\)

Step-by-Step Solution

Answer

The image curve is a line segment on the real axis from -2 to 2.

Step 1: Understand the Complex Mapping

The given complex mapping is \(w = z + \frac{1}{z}\). This means for any complex number \(z\), its image \(w\) is determined by this function. The goal is to determine how this function transforms the curve \(|z|=1\), which is a unit circle in the complex plane.

Step 2: Analyze the Mapping on the Unit Circle

For \(z\) on the unit circle, \(|z| = 1\), it can be represented as \(z = e^{i\theta}\) where \(\theta\) ranges from 0 to \(2\pi\). Thus \(w = z + \frac{1}{z} = e^{i\theta} + \frac{1}{e^{i\theta}} = e^{i\theta} + e^{-i\theta}\).

Step 3: Simplify the Expression for \(w\)

Notice that \(w = e^{i\theta} + e^{-i\theta}\) can be simplified using Euler's formula, which gives \(w = 2 \cos \theta\). This means the real part of \(w\) is \(2\cos\theta\) and the imaginary part is 0.

Step 4: Interpret the Resulting Image Curve

The simplification \(w = 2\cos\theta\) indicates that \(w\) ranges over all real numbers from \(-2\) to \(2\) as \(\theta\) varies from \(0\) to \(2\pi\). Therefore, the image of the unit circle under the mapping \(w = z + \frac{1}{z}\) is the line segment on the real axis between \(-2\) and \(2\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in complex numbers and resides in the complex plane. It is defined as the set of all complex numbers whose absolute value, or modulus, is equal to 1. This can be represented mathematically as \[ |z| = 1 \] where \( z \) is a complex number. Any point on the unit circle can also be expressed using Euler's formula:\[ z = e^{i\theta} \] Here, \( \theta \) represents the angle that the complex number makes with the positive real axis, varying from 0 to \( 2\pi \). In the unit circle, these numbers revolve infinitely around the origin but stay a constant distance from it.

• The magnitude of any complex number on this circle is always 1.
• The unit circle is crucial in trigonometry and Fourier analysis because of its periodic nature.
• It facilitates the visualization and understanding of complex exponential functions.

Complex Plane

The complex plane is a two-dimensional plane, similar to a coordinate plane in geometry, but designed for complex numbers. Each complex number \( z \) is represented in the form:\[ z = a + bi \] where \( a \) is the real part and \( bi \) is the imaginary part. This plane has a horizontal axis (real axis) and a vertical axis (imaginary axis). The intersection of these axes is the origin, which corresponds to the complex number 0.

• The real part of the complex number moves along the x-axis.
• The imaginary part moves along the y-axis.
• The magnitude, or modulus, of a complex number is the distance from the origin to the point \( (a, b) \).
• The angle between this point and the positive x-axis is called the argument of the complex number.

Understanding the complex plane is key to mastering complex analysis and visualizing complex functions.

Imaginary Part

The imaginary part of a complex number is the coefficient of \( i \), the imaginary unit. If a complex number is expressed as \( a + bi \), then \( bi \) is the imaginary part. The imaginary unit \( i \) has the unique property that:\[ i^2 = -1 \] This property helps extend the real numbers into complex numbers, allowing for the solutions of equations like \( x^2 + 1 = 0 \). The imaginary part contributes to the position of a complex number in the complex plane, influencing its distance and angle from the origin.

• Along with the real part, it determines the magnitude and direction of a complex number.
• In polar form, the imaginary part impacts the angle, or phase, represented by the argument \( \theta \).
• The imaginary part is zero when plotting on the real axis.

The combination of real and imaginary parts allows for a deeper understanding of the nature of complex numbers.

Euler's Formula

Euler's Formula is a remarkable equation in mathematics that establishes a deep relationship between exponential and trigonometric functions:\[ e^{i\theta} = \cos\theta + i\sin\theta \]This formula is particularly useful in converting complex numbers from their rectangular form (\( a + bi \)) to their polar form, where the magnitude is given by the modulus of the complex number, and the angle \( \theta \) is the argument.

• Euler's Formula is essential for understanding rotations in the complex plane.
• It plays a critical role in signal processing and electrical engineering, aiding in the analysis of periodic waveforms.
• The formula beautifully links exponential functions with circular functions, simplifying multiplication and division of complex numbers.

By harnessing Euler's Formula, one can unlock the potential of complex analysis, making it simpler to handle complex transformations and mappings.