Problem 8 - 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. Circular arc \(r=2,0 \leq \theta \leq \pi / 2\), under \(w=z^{1 / 2}\)

Step-by-Step Solution

Answer

The image curve is a circular arc of radius \(\sqrt{2}\) from \(0\) to \(\pi/4\) in the \(w\)-plane.

Step 1: Understand the Mapping Function

The mapping function given is \( w = z^{1/2} \). This means that each point \( z \) in the complex \( z \)-plane is transformed by taking its square root to find the corresponding point \( w \) in the \( w \)-plane.

Step 2: Define the Given Curve

The curve in the \( z \)-plane is a circular arc defined by \( r = 2 \) and \( 0 \leq \theta \leq \pi / 2 \). In polar form, each point on this arc can be expressed as \( z = 2e^{i\theta} \).

Step 3: Apply the Mapping to the Curve

Substitute \( z = 2e^{i\theta} \) into the mapping function: \( w = (2e^{i\theta})^{1/2} = \sqrt{2} e^{i\theta/2} \).

Step 4: Describe the Image Curve in the w-plane

The image curve is described by \( w = \sqrt{2} e^{i\phi} \), where \( \phi = \theta/2 \) and \( 0 \leq \theta \leq \pi/2 \). Consequently, \( 0 \leq \phi \leq \pi/4 \). This forms a circular arc in the \( w \)-plane with radius \( \sqrt{2} \) from \( \phi = 0 \) to \( \phi = \pi/4 \).

Key Concepts

Complex Mapping

Complex mapping involves transforming points from one complex plane to another using a specific function. In our exercise, the function used is \( w = z^{1/2} \), which indicates that we're taking the square root of each complex number \( z \).
This complex mapping converts each point \( z \) in the \( z \)-plane into a point \( w \) in the \( w \)-plane.To visualize this, consider the complex number \( z \) represented in terms of its magnitude \( r \) and angle \( \theta \) (in polar coordinates). The mapping \( w = z^{1/2} \) then calculates \( w \) as:

• The magnitude of \( w \) being the square root of \( r \).
• The angle of \( w \) being half of \( \theta \).

This transformation can cause shapes and angles to change. A circle in one plane may appear as a different shape in the other, revealing surprising yet beautiful results of complex mappings.

Polar Coordinates

Polar coordinates are an alternative coordinate system used to express complex numbers. Instead of using the standard \( (x, y) \) Cartesian coordinates, polar coordinates use a radius \( r \) and an angle \( \theta \). This is particularly useful in complex analysis when dealing with circles and rotations.
In polar form, complex numbers are expressed as \( z = r e^{i\theta} \), where \( e^{i\theta} \) represents a unit circle. The radius \( r \) gives the distance from the origin, while \( \theta \) gives the angle measured from the positive real axis. For the given curve in the exercise \( r = 2 \) and \( 0 \leq \theta \leq \pi / 2 \), these polar coordinates make describing arcs very intuitive as they relate directly to the geometry of the circles and arcs in the \( z \)-plane.
Using polar coordinates simplifies the process of applying functions like \( w = z^{1/2} \) when you can simply adjust the magnitude and the angle separately.

Circular Arc

In the context of complex analysis, a circular arc is a segment of a circle defined by a central angle. For our exercise, the arc in the \( z \)-plane is defined by a fixed radius \( r = 2 \) and angles ranging from \( \theta = 0 \) to \( \theta = \pi/2 \).
When this circular arc undergoes the mapping \( w = z^{1/2} \), its configuration in the \( w \)-plane transforms. The image arc retains the properties of being an arc but changes in size and angle. It now has a radius \( \sqrt{2} \) due to the magnitude transformation and spans a smaller angle \( 0 \leq \phi \leq \pi/4 \) because the function halves the angle.
This transformation of arcs is crucial in complex analysis and helps in visualizing complex functions. Understanding how arcs change under transformations can illuminate the nature of the function and assist in solving more complex problems.