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Chapter 1: Problem 28

Let \(n>2\) be an integer. Find the number of solutions to the equation $$ z^{n-1}=i \bar{z} $$

Short Answer

Expert verified
Answer: The equation has \(n-1\) distinct solutions.

Step by step solution

01

Rewriting the equation using polar form

Since we are dealing with a complex number equation, it is often helpful to rewrite complex numbers in polar form. First, multiply both sides of the equation by \(z\): $$ z^{n-1}\cdot z = i \bar{z} \cdot z $$ Which simplifies to: $$ z^n = iz. $$ Now let \(z=re^{i\theta}\), where \(r\) is the magnitude of \(z\) and \(\theta\) is the angle of \(z\) in polar coordinates. Replace \(z\) with its polar form in the equation: $$ (re^{i\theta})^n = i(re^{i\theta}). $$
02

Apply De Moivre's theorem

De Moivre's theorem states that \((re^{i\theta})^n = r^n e^{in\theta}\) for any positive integer \(n\). Apply this theorem to the left side of the equation: $$ r^n e^{in\theta} = i(re^{i\theta}). $$
03

Separate the magnitude and the angle

To proceed further, separate the magnitude and the angle parts of the equation: $$ r^n = |i(re^{i\theta})| \qquad \text{and} \qquad in\theta = Arg(i(re^{i\theta})). $$ The magnitude part simplifies to: $$ r^n = r, $$ which implies that \(r=1\) since \(n>2\). Next, consider the angle part: $$ in\theta = Arg(i(re^{i\theta})). $$ Since \(r=1\), \(in\theta = Arg(ie^{i\theta})=Arg(ie^{i(\theta+\frac{\pi}{2})})\). Now, we can find the solutions for the angle by considering \(Arg(z^n)=nArg(z)\): $$ n\left(\theta+\frac{\pi}{2}\right) = Arg(ie^{i\theta}) $$
04

Solve for the angle

Our goal is to find the number of solutions to the equation. For this, let's consider all \(k\in\mathbb{Z}\): $$ n\left(\theta+\frac{\pi}{2}\right)=2\pi k+\theta. $$ Rearrange the equation: $$ \theta(n-1)=2\pi k-\frac{n\pi}{2}. $$ Solving for \(\theta\): $$ \theta = \frac{2\pi k-\frac{n\pi}{2}}{n-1}. $$ Notice that \(\theta\) will have distinct values for all \(0\leq k\leq n-2\). Therefore, there are \(n-1\) distinct solutions in the equation. So, the number of solutions to the equation is \(n-1\).

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