Product of Complex Numbers in Polar Form/General Result

Theorem
Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.

Let $z_j = \left\langle{r_j, \theta_j}\right\rangle$ be $z_j$ expressed in polar form for each $j \in \left\{{1, 2, \ldots, n}\right\}$.

Then:
 * $z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \left({\cos \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right) + i \sin \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right)}\right)$

Proof
Proof by induction:

For all $n \in \N_{>0}$, let $P \left({n}\right)$ be the proposition:
 * $z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \left({\cos \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right) + i \sin \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right)}\right)$

Let this be expressed as:
 * $\displaystyle \prod_{j \mathop = 1}^n z_j = \prod_{j \mathop = 1}^n r_j \sum_{j \mathop = 1}^n \left({\cos \theta_j + i \sin \theta_j}\right)$

$P \left({1}\right)$ is the case:


 * $r_1 \left({\cos x + i \sin x}\right)^1 = r_1 \left({\cos \left({1 x}\right) + i \sin \left({1 x}\right)}\right)$

which is trivially true.

Basis for the Induction
$P \left({2}\right)$ is the case:


 * $z_1 z_2 = r_1 r_2 \left({\cos \left({\theta_1 + \theta_2}\right) + i \sin \left({\theta_1 + \theta_2}\right)}\right)$

which is proved in Product of Complex Numbers in Polar Form.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle \prod_{j \mathop = 1}^k z_j = \prod_{j \mathop = 1}^k r_j \sum_{j \mathop = 1}^k \left({\cos \theta_j + i \sin \theta_j}\right)$

Then we need to show:
 * $\displaystyle \prod_{j \mathop = 1}^{k+1} z_j = \prod_{j \mathop = 1}^{k+1} r_j \sum_{j \mathop = 1}^{k+1} \left({\cos \theta_j + i \sin \theta_j}\right)$

Induction Step
This is our induction step:

Hence, by induction, for all $n \in \N_{>0}$:


 * $z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \left({\cos \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right) + i \sin \left({\theta_1 + \theta_2 + \cdots + \theta_n}\right)}\right)$