Problem 10 - Advanced Engineering Mathematics

Question

Find the image of the region \(|z| \leq 1\) in the upper half-plane under the complex mapping \(w=z+1 / z\). What is the image of the line segment \([-1,1]\) on the \(x\)-axis?

Step-by-Step Solution

Answer

The image of \(|z| \leq 1\) in \(\text{Im}(z) > 0\) is an ellipse \(\frac{x^2}{4} + y^2 = 1\); segment \([-1,1]\) maps to \([-2,2]\).

Step 1: Understanding the Mapping

The mapping given is \( w = z + \frac{1}{z} \). Our goal is to find the image of the region \(|z| \leq 1\) in the upper half-plane under this mapping. Essentially, we need to determine how points within the unit circle transform under this function.

Step 2: Mapping the Boundary

Start with the boundary \(|z| = 1\), which is the unit circle. Substitute \( z = e^{i\theta} \): for \( \theta \) ranging from \( 0 \) to \( \pi \) (since we're focusing on the upper half-plane). Therefore, \( w = e^{i\theta} + e^{-i\theta} = 2\cos(\theta) \), which maps to the real segment \([-2, 2]\) on the real axis since \(\cos(\theta)\) ranges from -1 to 1.

Step 3: Mapping the Interior

Consider a point \( z = re^{i\theta} \) where \( 0 \leq r \leq 1 \) and \( 0 \leq \theta \leq \pi \). The mapping becomes \( w = re^{i\theta} + \frac{1}{r}e^{-i\theta} = r\cos(\theta) + i r\sin(\theta) + \frac{1}{r}\cos(\theta) - i \frac{1}{r} \sin(\theta) \). Simplifying, the real part is \( \left( r + \frac{1}{r} \right) \cos(\theta) \) and the imaginary part is \( \left( r - \frac{1}{r} \right) \sin(\theta) \), creating a region within the ellipse defined by \( \frac{x^2}{4} + \frac{y^2}{1} = 1 \).

Step 4: Mapping the Line Segment

To find the image of the line segment \([-1,1]\) on the \(x\)-axis, consider \( z = x \) where \(-1 \leq x \leq 1\). Substitute \( z \) in the mapping: \( w = x + \frac{1}{x} \). This transformation maps the interval \([-1, 1]\) to the segment on the real axis between \([-2, 2]\).

Step 5: Constructing the Image

The boundary of the transformed region from \(|z| \leq 1\) is mapped onto an ellipse in the \(w\)-plane. It includes the segment \([-2, 2]\) in the real axis. The upper half-plane (the circle interior) maps smoothly onto this elliptical boundary, emphasizing that the entire region is now within the ellipse \(\frac{x^2}{4} + y^2 = 1\).

Key Concepts

Conformal Mapping

Conformal mappings are fascinating mathematical tools in complex analysis. They are functions that locally preserve angles, which means they map small shapes to other small shapes without distorting angles. This property makes conformal mappings particularly useful in physics and engineering. They help transform complicated domains into simpler ones.

For the mapping \( w = z + \frac{1}{z} \), our task involves transforming a region in the complex plane. The given function will map the region inside the unit circle into a new shape in the complex \(w\)-plane. This transformation is both angle-preserving and bijective within the specified region, ensuring that no overlaps occur. Such transformations are widely used to simplify the complex geometrical shapes into more manageable ones while keeping critical features such as angle relations intact.

Unit Circle

The unit circle is a straightforward yet fundamental concept in complex analysis. It is a set of points described by \( |z| = 1 \), where every point on the circle has a magnitude of 1. In the complex plane, it corresponds to the boundary of a disk centered at the origin.

In the context of our exercise, we focus on mapping the unit circle, especially its upper half, to observe how it transforms under the mapping. Representing the points on this circle as \( z = e^{i\theta} \), with \( \theta \) ranging from 0 to \( \pi \) for the upper half, allows us to discover that the mapping \( w = z + \frac{1}{z} \) converts these points into the real interval from -2 to 2. Thus, the unit circle's upper portion beautifully maps onto a real line segment due to the simplicity of cosine terms \( 2\cos(\theta) \).

Upper Half-Plane

When dealing with the upper half-plane, we are examining all points in the complex plane with positive imaginary parts. It's a common setting in complex analysis for establishing mappings since it often simplifies infinite extent geometries into finite mappings.

For this example, we're exploring how the region within the unit circle, specifically the upper half portion, maps through the function \( w = z + \frac{1}{z} \). The transformation of points inside the unit circle, characterized by \( z = re^{i\theta} \) where \( 0 \leq r \leq 1 \) and \( 0 \leq \theta \leq \pi \), leads to a mapped region that takes the shape of an ellipse in the \(w\)-plane. This transformation accounts for both the real and imaginary components, resulting in a beautifully bounded elliptical region.

Ellipse Mapping

Ellipse mapping is an elegant outcome of certain complex transformations, such as our mapping function \( w = z + \frac{1}{z} \). An ellipse in the complex plane is the image of a circle under this transformation.

To understand why we obtain an ellipse, we look at the real and imaginary parts of the transformed points inside the unit circle. For the function at hand, the real part, \( ( r + \frac{1}{r} ) \cos(\theta) \), and the imaginary part, \( ( r - \frac{1}{r} ) \sin(\theta) \), together satisfy the elliptical condition. This leads to regions confined within the ellipse \( \frac{x^2}{4} + y^2 = 1 \), as derived in the mapping. As you can see, the real axis, representing the major axis of the ellipse, stretches from -2 to 2, forming a bounded and symmetric shape typical of elliptical mappings.