Problem 5 - Advanced Engineering Mathematics

Question

For \(w=\operatorname{Ln} z, u=\log _{e}|z|\) and \(v=\operatorname{Arg} z .\) The semi-circle \(|z|=1, y>0\) may also be described by \(r=1\) \(0<\theta<\pi .\) Therefore \(u=0\) and \(0

Step-by-Step Solution

Answer

The image is a vertical line segment from 0 to \\pi i\\.

Step 1: Understanding the Problem

Our goal is to describe the mapping of a semicircle \( |z| = 1, y > 0 \) in the complex plane under the complex logarithm function. We begin by identifying the transformation using \( w = ext{Ln} z \), where \( w = u + iv \) with \( u = ext{log}_e |z| \) and \( v = ext{Arg} z \).

Step 2: Identify Parameter Values for Semicircle

Given \( |z| = 1 \), this implies \( u = ext{log}_e |z| = ext{log}_e 1 = 0 \). Additionally, the semicircle bounds are \( 0 < heta < \pi \), indicating \( 0 < v < \pi \).

Step 3: Describe the Transformation

With \( u = 0 \) and the range \( 0 < v < \pi \), the transform maps onto the vertical line where \( w = iv \). As \( v \) varies from 0 to \pi\, the path in the complex plane is described by the line segment from \( w = 0 \) to \ (w = \pi i) \.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to represent complex numbers. Each complex number has a real part (x-axis) and an imaginary part (y-axis). This plane is crucial for visualizing the behavior and transformations of complex functions.

Let's break it down:

• **Real part of a complex number**: This is represented by the x-coordinate on the complex plane.
• **Imaginary part**: This is depicted as the y-coordinate and is denoted with an 'i' (imaginary unit).
• **Polar coordinates**: A complex number can also be expressed in polar form where the modulus (|z|) represents the distance from the origin and the argument (angle \(\theta\)) represents the direction.

The complex plane allows us to see the effects of transformations and understand mappings like those performed by functions such as the complex logarithm. In our exercise, a semicircle represented by \(|z| = 1\) is mapped onto a segment in this plane, providing insight into how complex functions interact with geometric figures.

Complex Logarithm

The complex logarithm is an extension of the logarithm function to complex numbers. It involves more complexity due to the nature of the complex plane. Let's dig into what it involves:

• The logarithm of a complex number \( z = re^{i\theta} \) (in polar form) is given by \( \operatorname{Ln}(z) = \ln(r) + i\theta \).
• **Modulus** \( |z| \): For the complex logarithm, the natural logarithm of the modulus becomes the real part.
• **Argument** \( \operatorname{Arg}(z) \): This is the imaginary part, accounting for the angle in the polar coordinate system.
• **Branch cuts**: Due to its multi-valued nature, the complex logarithm involves a principal value determined by establishing branch cuts, typically along the negative real axis.

In our exercise's context, the semicircle with \(|z| = 1\) allows \( \ln|z| = 0 \) and creates a vertical line in the complex plane as the imaginary argument \(v\) varies from \(0\) to \(\pi\). This illustrates the nuanced transformation applied to geometric figures via the logarithmic mapping.

Argand Diagram

The Argand diagram is a method to represent complex numbers graphically on the complex plane. Named after Jean-Robert Argand, it's an essential tool for visualizing how complex numbers behave under various transformations.

Key concepts:

• **Coordinates**: Argand diagrams map real numbers to the horizontal axis and imaginary numbers to the vertical axis.
• **Semicircle representation**: In our exercise, \(|z| = 1, y > 0\) describes the upper semicircle, translating to angles between \(0\) and \(\pi\) radians.
• **Visual transformation**: In the Argand diagram, our semicircle is transformed into a line segment from \(w = 0\) to \(w = \pi i\).

This form of visualization is particularly handy when dealing with complex transformations, as it allows one to see clearly how different values move and change location on the plane. Such visual tools simplify the understanding of the relationship and transformation of complex functions, providing a visual insight into abstract mathematical concepts.