Problem 5 - Complex Analysis for Mathematics and Engineering

Question

Find a parametrization of the curve that is a portion of the circle \(|z|=1\) that joins the point \(-i\) to \(i\) if (a) the curve is the right semicircle. (b) the curve is the left semicircle.

Step-by-Step Solution

Answer

(a) \(z(t) = e^{it}\), \(t \in [-\pi/2, \pi/2]\); (b) \(z(t) = e^{it}\), \(t \in [\pi/2, 3\pi/2]\).

Step 1: Understand the Circle

The equation \(|z| = 1\) describes a circle of radius 1 centered at the origin in the complex plane. Any point \(z\) on this circle can be represented as \(z = e^{i\theta}\) due to its unit magnitude.

Step 2: Parametrize the Right Semicircle

To describe the right semicircle, the angle \(\theta\) should vary from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). The parametric equation for this semicircle is \(z(t) = e^{it}\) where \(t\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 3: Parametrize the Left Semicircle

For the left semicircle joining \(-i\) to \(i\), \(\theta\) should vary from \(\frac{\pi}{2}\) to \(-\frac{\pi}{2}\). The parametric equation in this case is \(z(t) = e^{it}\) with \(t\) ranging from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\), or equivalently by making a transition from right to left.

Key Concepts

Parametrization

Parametrization is a technique used to express a curve in terms of a single variable, usually denoted by a letter like \( t \) or \( \theta \). This makes it easier to analyze and work with curves, especially in complex spaces like the complex plane.
For example, any curve on a plane can be described by giving a set of equations that express all coordinates of the points on the curve.

• In simple terms, it's like providing a recipe to trace the curve point by point.
• This is essential when dealing with problems in calculus and complex analysis.

For instance, to find the parametric equation for a circle, we use trigonometric functions. On the unit circle, any point \( z \) can be expressed as \( z = e^{i\theta} \), where \( \theta \) varies over a specific range. Thus, making parametrization an indispensable tool for visualizing and working with curves in the complex plane.

Complex Plane

The complex plane is a two-dimensional plane used to represent complex numbers visually. Imagine it's like the traditional Cartesian coordinate system, but where one axis represents real numbers, and the other represents imaginary numbers.
Here's a break down of its main components:

• The horizontal axis is called the real axis.
• The vertical axis is known as the imaginary axis.

Complex numbers are represented as points or vectors in this plane using a format like \( z = x + yi \) where \( x \) is the real part, and \( yi \) is the imaginary part.
When dealing with complex functions, visualizing transformations and behaviors becomes straightforward on the complex plane. For example, the circle \( |z| = 1 \) describes all points at a unit distance from the origin, a feature easily modelled within this framework. Parametrization on the complex plane allows us to express intricate paths and shapes succinctly.

Unit Circle

The unit circle in complex analysis is a special circle in the complex plane with a radius of one, centered at the origin. It holds a key position in both trigonometry and complex analysis because of its unique properties.
A few significant traits are:

• Every point \( z \) on the unit circle satisfies the equation \( |z| = 1 \), meaning it has a magnitude of one.
• It serves as the stage for defining and exploring complex numbers through exponentials, e.g., \( z = e^{i\theta} \).
• It facilitates the visualization of complex roots of unity, which are points evenly spaced along the circle.

Given its mathematical significance, many problems involve parametrizing sections of the unit circle, like the right or left semicircle, making it an essential concept to grasp. Parametrization helps us understand how complex numbers and their magnitudes interact geometrically.