Properties of Complex Numbers in Polar Form

For a complex number in rectangular form, $z = a + ib$, its complex conjugate is defined as $\bar{z} = a - ib$.

Sometimes, instead of $\bar{z}$, we will denote the complex conjugate as $z^{*}$. Both notations are used frequently in the mathematical sciences.

These are completely equivalent notations, and just personal preference whether we use $\bar{z}$ or $z^{*}$ to refer to the complex conjugate of a complex number $z$.

For a complex number in polar form, $z = |z|e^{i\theta}$, what is its complex conjugate?

First, consider just $e^{i\theta}$, with complex conjugate denoted by $\left(e^{i\theta}\right)^*$. Using Euler's formula:

$$\begin{align*} \left(e^{i\theta}\right)^* &= \left(\cos(\theta) + i\sin(\theta)\right)^* \\ &= \cos(\theta) - i\sin(\theta) \\ &= \cos(-\theta) + i\sin(-\theta) \\ &= e^{-i\theta} \end{align*}$$

because of the trigonometric identities: $\cos(-\theta) = \cos(\theta)$ and $\sin(-\theta) = -\sin(\theta)$.

Thus, given a complex number $z = |z|e^{i\theta}$, its complex conjugate, $z^*$, is:

$$\begin{align*} z^* &= \left(|z|e^{i\theta}\right)^* \\ &= |z|^* \left(e^{i\theta}\right)^* \\ &= |z| e^{-i\theta} \\ \end{align*}$$

since $|z|$ is always a real number, taking the complex conjugate of $|z|$ leaves it unchanged.

Furthermore, notice that:

\begin{align*} |z| &= \sqrt{z^*z} \\ ...