How can I express in polar form $z = \frac{1}{(x-3 + iy)^n}$?

Question

How can I express in polar form

$$z = \frac{1}{(x-3 + iy)^n} $$

$x$, $y$ and $n$ could be any number. $i$ is the imaginary unit.

Answer 1

We first compute the polar form of $a = x-3+iy$, giving $r = \sqrt {(x-3)^2 + y^2}$ and $\theta = \tan^{-1} \left( \frac{y}{x-3} \right)$. Writing it as $re^{i \theta}$ and noting that $z = a^{-n}$, we obtain the polar form $r^{-n} e^{-in \theta}$ where $r$ and $\theta$ are the aforementioned constants in terms of $x$ and $y$.