- Complex n-th roots.
- Complex exponential.
Complex N-th Roots
Given $$ \begin{align} n \in \mathbb{N}, z \in \mathbb{C} \ \text{find}\ w\ \text{such that}\ w^n = Z \end{align}
- We start by converting the two into polar form. - Since $w^n = z$, we can say the same about their magnitudes. $|w|^n = |z|^n$. - We have a similar expression for the angles of the two numbers.\begin{align} n\phi = \theta + 2\pi k \ \phi = \frac{\theta + 2\pi k}{n} \end{align}
For $k \in [0, n)$ as there will be $n$ solutions to $w^n = z$. We limit $k$ to the interval as after that we'll continue getting repeating values (the angle is going around a circle) # Complex Exponential To us, for now it is just a more concise notation. I should watch the 3b1b video to understand the meaning behind this.\begin{align} e^{j\theta} = \cos\theta+j\sin\theta\ \implies z = r\cdot e^{j\theta} \end{align}