Problem 16 - Advanced Engineering Mathematics

Question

If \(w=2 z-1\) and \(|z|=1,\) then, since \(z=\frac{w+1}{2},\left|\frac{w+1}{2}\right|=1\) or \(|w+1|=2 .\) The image is a circle with center at \(w=-1\) and with radius \(r=2.\)

Step-by-Step Solution

Answer

The image is a circle with center at \(w=-1\) and radius \(r=2\).

Step 1: Understanding the given equation

We are given that the modulus of \(z\) is 1, i.e., \(|z|=1\). The equation for \(w\) in terms of \(z\) is \(w = 2z - 1\). We also have the expression for \(z\) as \(z = \frac{w + 1}{2}\). This substitution will help us find the relation between \(w\) and \(z\).

Step 2: Substituting in the modulus condition

Given \(|z| = 1\), we substitute \(z = \frac{w + 1}{2}\) into the modulus condition: \[ \left| \frac{w + 1}{2} \right| = 1 \].

Step 3: Solving the modulus equation

The equation \( \left| \frac{w + 1}{2} \right| = 1 \) simplifies to \( \frac{|w + 1|}{2} = 1 \). Solving for \(|w + 1|\) gives us \(|w + 1| = 2\).

Step 4: Describing the geometric shape

The equation \(|w + 1| = 2\) represents a circle in the complex plane. The center of this circle is at \(w = -1\), and the radius of the circle is 2.

Key Concepts

Complex Modulus

In complex analysis, the modulus of a complex number is a crucial concept, helping to measure the 'size' or 'magnitude' of the number. For a complex number represented as \( z = a + bi \) (where \( i \) is the imaginary unit such that \( i^2 = -1 \)), its modulus is calculated using the formula: \[|z| = \sqrt{a^2 + b^2}.\] The modulus gives us the distance of the complex number from the origin on the complex plane. This concept is key when examining complex numbers' geometric properties and performing transformations within the plane.

• The modulus always results in a non-negative real number.
• It is analogous to the absolute value for real numbers.
• Used widely for defining other geometric properties like distance and angle.

Complex Plane Geometry

The complex plane, also known as the Argand plane, is a two-dimensional plane representing complex numbers. It combines real and imaginary components as coordinates: real part on the x-axis and imaginary part on the y-axis. This geometric representation enables us to easily visualize complex number operations such as addition and multiplication. The scope of complex plane geometry includes various transformations and symmetries, making it a fundamental part of understanding complex numbers. Besides representing numbers, the plane is essential for visualizing complex functions and their behaviors.

• Points correspond to complex numbers.
• Operations like addition can be visualized as vector addition.
• Transformations can depict rotations and dilations.

When dealing with geometric shapes in the complex plane, such as circles or lines, the concepts of modulus and arguments often come into play, providing deeper insights into their properties and interrelations.

Circle Equation in Complex Plane

A circle in the complex plane can likewise be defined much like in Cartesian geometry but using complex modulus properties. When given an equation of the form \(|z - a| = r\), it represents a circle with:

• Center at the complex point \(a\).
• Radius of length \(r\).

In the context of our problem, the equation \(|w + 1| = 2\) denotes a circle centered at \(w = -1\) and having a radius of 2 in the complex plane. This arises from the equation transformation process derived using the modulus property. Such representations using complex numbers provide a succinct and powerful way to handle and interpret geometric figures. Moreover, they emphasize the harmony between algebraic expressions and geometric interpretations in complex analysis. Understanding these formulations supports solving complex equations through geometric methods, offering another perspective for identifying solutions.

• Utilizes the distance from a fixed point to describe the locus of points forming a circle.
• Reveals connections between complex modulus and Euclidean distance.