Find all the values of $w$ ∈ C that satisfy the equation.

Question

Find all values of $\omega \in \mathbb{C}$ such that $\frac{\omega - \frac1\omega}{2i} = 2.$

So what I have is:

Let $\omega$ be represented as $x + yi$ (by definition of complex numbers)

Then $\frac1\omega = \frac{x-yi}{x^2 + y^2}$ (by properties of complex numbers)

So, $\frac{x+yi - \frac{x-yi}{x^2 + y^2}}{2i} = 2$

Eventually, I get, via expansion and simplification:

$\frac{x(x^2 + y^2 - 1) + yi(x^2 + y^2 + 1)}{2i(x^2 + y^2)}$

Any hints on how I can continue? I feel like I'm close but missing something.

Answer 1

Note that

$$\frac{\omega - \frac1\omega}{2i} = 2\iff \omega - \frac1\omega=4i \stackrel{\omega \ne 0}{\iff} \omega^2-1=4i\omega\iff\omega^2-4i\omega-1=0.$$

This equation should be much more familiar.

Answer 2

This is an ordinary algebraic equation in the complex variable $\omega$, and in cases like this, it’s almost always not wise to look at the real and imaginary parts of $\omega$.

Just pretend you’re in high-school and solve: \begin{align} \omega + 1/\omega&=4i\\ \omega^2-4i\omega-1&=0\\ \omega=2i\pm\sqrt{-3}\\ \omega=(2\pm\sqrt3)i \end{align}