Problem 5 - 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. Semicircle \(|z|=1, y>0\), under \(w=\operatorname{Ln} z\)

Step-by-Step Solution

Answer

The image curve is the segment of the imaginary axis from 0 to \(i\pi\).

Step 1: Understand the given semicircle in the z-plane

The semicircle is given by the equation \(|z| = 1\), with the condition that \(y > 0\). This represents the upper half (since \(y > 0\) refers to the imaginary part being positive) of the unit circle in the complex plane. The unit circle is centered at the origin with a radius of 1.

Step 2: Define the mapping function

The mapping function is given by \(w = \operatorname{Ln} z\), which represents the natural logarithm of the complex number \(z\). The logarithm of a complex number \(z = re^{i\theta}\) is given by \(\operatorname{Ln} z = \ln r + i\theta\). For the unit circle where \(r = 1\), this simplifies to \(\operatorname{Ln} z = i\theta\).

Step 3: Set the constraints according to the mapping

Since the semicircle \(|z| = 1\) is described by \(z = e^{i\theta}\) for \(y > 0\), \(\theta\) will range from 0 to \(\pi\). Thus, substituting this into \(w = \operatorname{Ln} z\), we find \(w = i\theta\) with \(0 < \theta < \pi\).

Step 4: Describe the image curve in the w-plane

The image curve is based on \(w = i\theta\) for \(0 < \theta < \pi\). This expression represents the imaginary axis segment from 0 to \(i\pi\). This is the image of the upper half of the unit circle under the mapping \(w = \operatorname{Ln} z\).

Key Concepts

Complex Mapping

Complex mapping is an important concept in complex analysis, focusing on how one complex plane transforms into another. When dealing with complex mappings, we often have a function that takes a complex number from one plane (often called the \(z\)-plane) and maps it to another complex number on a different plane (called the \(w\)-plane). This transformation can change shapes, sizes, and positions of geometric figures.

• Mapping can illustrate various configurations, such as circles transforming into lines or even different circles.
• By studying complex mappings, we can better understand the properties and behaviors of complex functions.

In the exercise, the semicircle \(|z| = 1, y > 0\) is mapped using the function \(w = \operatorname{Ln} z\). Here, our task is to determine how the upper half of the unit circle transforms in the \(w\)-plane. A crucial part of understanding mappings is knowing the characteristics of the function used, which brings us to the natural logarithm of complex numbers.

Natural Logarithm of Complex Numbers

The natural logarithm of complex numbers extends the familiar logarithm concept from real numbers to complex ones. For a complex number \(z\) expressed in polar form as \(z = re^{i\theta}\), the natural logarithm is given by \(\operatorname{Ln} z = \ln r + i\theta\). This breaks down as follows:

• \(\ln r\) represents the logarithm of the magnitude (modulus) of the complex number.
• \(i\theta\) represents the argument (angle) of the complex number, where \(\theta\) is measured in radians.

In our specific problem, the modulus \(r\) of the semicircle is 1, implying \(\ln r = 0\). Therefore, the natural logarithm simplifies to \(\operatorname{Ln} z = i\theta\). Understanding this helps us map points from the semicircle to the \(w\)-plane effectively, as the resulting transformation becomes directly dependent on the angle \(\theta\).

Image Curves in Complex Plane

The concept of image curves in the complex plane focuses on the resulting figures after applying a complex mapping function. In the present exercise, we determine that the mapping \(w = \operatorname{Ln} z\) translates the upper semicircle into a segment of the imaginary axis.

• The semicircle \(|z| = 1\) implies each point is \(z = e^{i\theta}\).
• Substituting \(z\) into \(w = \operatorname{Ln} z\) gives \(w = i\theta\).
• Since \(\theta\) ranges from 0 to \(\pi\) (due to \(y > 0\)), the resulting image curve is the line segment from 0 to \(i\pi\) on the imaginary axis.

This paints a vivid picture of how the complex mapping has transformed the semicircle into a new geometric shape on the \(w\)-plane. Understanding these transformations aids in visualizing the behavior and characteristics of complex numbers under various mappings.