Sum of Complex Numbers in Exponential Form/General Result

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Theorem

Let $n \in \Z_{>0}$ be a positive integer.

For all $k \in \set {1, 2, \dotsc, n}$, let:

$z_k = r_k e^{i \theta_k}$


Let:

$r e^{i \theta} = \displaystyle \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$


Then:

\(\displaystyle r\) \(=\) \(\displaystyle \sqrt { {r_1}^2 + {r_2}^2 + \dotsb + {r_n}^2 + \ldots}\)
\(\displaystyle \theta\) \(=\) \(\displaystyle \map \arctan {\dfrac {r_1 \sin \theta_1 + r_1 \sin \theta_2 + \dotsb + r_n \sin \theta_2} {r_1 \cos \theta_1 + r_1 \cos \theta_2 + \dotsb + r_n \cos \theta_2} }\)


Proof


Sources