Problem 10
Question
Circle \(|z|=1\) under \(w=z+1 / z\)
Step-by-Step Solution
Verified Answer
The transformation maps the unit circle to the interval \([-2, 2]\) on the real axis.
1Step 1: Understanding the Given Circle
The given circle is defined by the equation \(|z| = 1\). This means any complex number \(z = x + yi\) that lies on this circle satisfies the equation \(x^2 + y^2 = 1\). Essentially, this circle is the unit circle in the complex plane centered at the origin.
2Step 2: Substitute and Simplify the Transformation
The transformation given is \(w = z + \frac{1}{z}\). Let's substitute \(z = e^{i\theta}\) for \(z\) on the unit circle (since \(|z| = 1\), a complex number on the unit circle can be expressed in this form). Then \(\frac{1}{z} = e^{-i\theta}\). Thus, the transformation becomes:\[ w = e^{i\theta} + e^{-i\theta} = 2\cos\theta \]
3Step 3: Analyzing the Resulting Shape in the w-plane
From the simplification, we have \(w = 2\cos\theta\), where \(\theta\) ranges from 0 to \(2\pi\). The range of \(\cos\theta\) is from -1 to 1, which means \(w\) ranges from -2 to 2 along the real line. Thus, the transformation maps the unit circle to the interval \([-2, 2]\) on the real axis of the \(w\)-plane.
Key Concepts
Complex NumbersUnit CircleComplex Plane
Complex Numbers
Complex numbers are like the superheroes of the number world. They are written in the form of \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit. The value of \(i\) is defined as \( \sqrt{-1} \), which is why it is called 'imaginary.'
Complex numbers extend our number system. By including them, we can solve equations that do not have real solutions. For example, the equation \(x^2 + 1 = 0\) has no solution in the real numbers, but in the complex numbers, it does: \(x = i\) and \(x = -i\).
Why are they so special? Here's a simple way of understanding it:
Complex numbers extend our number system. By including them, we can solve equations that do not have real solutions. For example, the equation \(x^2 + 1 = 0\) has no solution in the real numbers, but in the complex numbers, it does: \(x = i\) and \(x = -i\).
Why are they so special? Here's a simple way of understanding it:
- The real part of a complex number \(z\) is the "\(x\)" in \(z = x + yi\).
- The imaginary part is the "\(y\)" in \(z = x + yi\).
- We can plot complex numbers on a plane, called the complex plane, using both \(x\) (real axis) and \(y\) (imaginary axis).
Unit Circle
The unit circle is an important concept in both mathematics and complex analysis. It is basically a circle with a radius of 1, centered at the origin in the complex plane.
When we say a complex number \(z\) lies on the unit circle, it satisfies \(|z| = 1\) or \(z = e^{i\theta}\). Here, \(\theta\) is an angle which describes the position of \(z\) relative to the positive real axis. Imagine walking around the edge of a circle with every step changing \(\theta\). By varying \(\theta\) from 0 to \(2\pi\), every point on the unit circle can be described. This makes it a powerful tool to study periodic phenomena in the complex plane.
Notice how we expressed \(z\) using Euler's formula \(e^{i\theta}\), which is very common with unit circles:
When we say a complex number \(z\) lies on the unit circle, it satisfies \(|z| = 1\) or \(z = e^{i\theta}\). Here, \(\theta\) is an angle which describes the position of \(z\) relative to the positive real axis. Imagine walking around the edge of a circle with every step changing \(\theta\). By varying \(\theta\) from 0 to \(2\pi\), every point on the unit circle can be described. This makes it a powerful tool to study periodic phenomena in the complex plane.
Notice how we expressed \(z\) using Euler's formula \(e^{i\theta}\), which is very common with unit circles:
- It shows the interconnectedness between trigonometry and complex analysis.
- It allows a more elegant way to express complex transformations, as seen in this exercise.
Complex Plane
The complex plane is a two-dimensional plane used to visually represent complex numbers. Think of it as a map for complex numbers with a real part and an imaginary part. The horizontal axis is called the real axis, and the vertical axis is the imaginary axis.
In this plane, any complex number \(z = x + yi\) can be represented as a point where \(x\) is the position along the real axis, and \(y\) is the position along the imaginary axis. This makes understanding and visualizing operations on complex numbers much easier:
The complex plane is especially useful in visualizing such transformations and comprehending their impacts on complex analysis problems.
In this plane, any complex number \(z = x + yi\) can be represented as a point where \(x\) is the position along the real axis, and \(y\) is the position along the imaginary axis. This makes understanding and visualizing operations on complex numbers much easier:
- Adding complex numbers becomes as simple as vector addition in this plane.
- Multiplying them involves rotating and scaling, providing rich geometrical insight.
The complex plane is especially useful in visualizing such transformations and comprehending their impacts on complex analysis problems.
Other exercises in this chapter
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